Unbounded Hamiltonians generated by Parseval frames

Abstract

In Bagarello and Kużel (Acta Appl Math 171:4, 2021) Parseval frames were used to define bounded Hamiltonians, both in finite and in infinite dimesional Hilbert spaces. Here we continue this analysis, with a particular focus on the discrete spectrum of Hamiltonian operators constructed as a weighted infinite sum of rank one operators defined by some Parseval frame living in an infinite dimensional Hilbert space. The main difference with Bagarello and Kużel (Acta Appl Math 171:4, 2021) is that, here, the operators we consider are mostly unbounded. This is an useful upgrade with respect to our previous results, since physically meaningful Hamiltonians are indeed often unbounded. However, due to the fact that frames (in general) are not bases, the definition of an Hamiltonian is not so easy, and part of our results goes in this direction. Also, we analyze the eigenvalues of the Hamiltonians, and we discuss some physical applications of our framework.

1 Introduction

In quantum mechanics a central object is the Hamiltonian H of the physical system ${{\mathcal {S}}}$ one is interested in. H is, for closed and conservative systems, the energy of ${{\mathcal {S}}}$. Quite often, the first step to analyze ${{\mathcal {S}}}$ consists in finding the eigenvectors of H, $e_j$: $H\,e_j=E_je_j$, $j=1,2,3,\ldots$. If H is self-adjoint, then each $E_j\in {\mathbb {R}}$ and eigenvectors related to different eigenvalues are orthogonal: $\left\langle e_j,e_k\right\rangle =0$, if $E_j\ne E_k$. Most of the times ${{\mathcal {F}}}_e=\left\{ e_j\right\}$ is an orthonormal basis (ONB) for $\mathcal{H}$, the Hilbert space where ${{\mathcal {S}}}$ is defined, and the Hamiltonian is written

$$\begin{aligned} H=\sum _jE_j\left\langle e_j,\cdot \right\rangle \,e_j=\sum _jE_jP_j, \end{aligned}$$

(1.1)

where we have introduced $P_j$ acting on a generic vector f of $\mathcal{H}$ as follows: $P_jf=\left\langle e_j,f\right\rangle \,e_j$. Since $P_j=P_j^*=P_j^2$, H can be seen as a weighted sum of orthogonal projectors, with weights given by its eigenvalues $E_j$. If is clear that the domain of H, $\mathcal {D}(H)$, is not necessarily all of $\mathcal{H}$. In fact we have $\mathcal {D}(H)=\left\{ f\in \mathcal{H}:\,\, \sum _j{E_j^2}|\left\langle e_j,f\right\rangle |^2<\infty \right\}$, which is surely dense in $\mathcal{H}$ (it contains the linear span of the $e_j$’s), but not necessarily coincident with $\mathcal{H}$.

In the past 30 years it has become clearer and clearer that the Hamiltonian of a system does not really have to be self-adjoint [13]. Since this seminal paper, an always increasing number of physicists and mathematicians started to consider this possibility, where the reality of the eigenvalues of a manifestly non self-adjoint Hamiltonian H is due to some physical symmetry rather than on the mathematical requirement that $H=H^*$.

However, even if the eigenvalues of H are real, the eigenvectors, $\varphi _j$ are no longer mutually orthogonal, in general. Still, if the set ${{\mathcal {F}}}_\varphi =\left\{ \varphi _j\right\}$ is a basis for $\mathcal{H}$, a second (uniquely determined) basis of the Hilbert space also exists, ${{\mathcal {F}}}_\psi =\left\{ \psi _j\right\}$, such that $\left\langle \varphi _j,\psi _k\right\rangle =\delta _{j,k}$, and $H^*\psi _k=\overline{E_k}\,\psi _k$. In this case we can write H and $H^*$ as follows:

$$\begin{aligned} H=\sum _jE_j\left\langle \psi _j,\cdot \right\rangle \,\varphi _j=\sum _jE_jQ_j, \qquad H^*=\sum _j\overline{E_j}\left\langle \varphi _j,\cdot \right\rangle \,\psi _j=\sum _j\overline{E_j}\,Q_j^*, \end{aligned}$$

where $Q_jf=\left\langle \psi _j,f\right\rangle \,\varphi _j$ is a projection operator, but it is not orthogonal if $\varphi _j\ne \psi _j$. Operators of this kind have been studied in the past, mainly from a mathematical point of view. We refer to [6] for some results, mainly for the case in which ${{\mathcal {F}}}_\varphi$ and ${{\mathcal {F}}}_\psi$ are Riesz bases. Later, generalizations of this situation have been considered [5, 18, 19]. In all these extensions, biorthogonality of the sets of vectors used to define some specific Hamiltonian was required. In [9], this assumption was removed, in our knowledge, for the first time: we used frames, and in particular Parseval frames (PF), rather than bases. Hence the existence of a biorhogonal basis is not guaranteed. Our interest was mainly mathematical, but was also based on a very simple physical remark: in the analysis of a concrete physical situation it may happen that not all vectors of $\mathcal{H}$ are relevant in the analysis of ${{\mathcal {S}}}$. For instance, when the energy of ${{\mathcal {S}}}$ cannot really increase too much, or when ${{\mathcal {S}}}$ is localized in a bounded region, or still when the value of the momentum of ${{\mathcal {S}}}$ cannot be too large. In all these cases, but not only, it is reasonable to consider a physical vector space, $\mathcal{H}_{ph}$, as the subset of the mathematical Hilbert space $\mathcal{H}$ on which ${{\mathcal {S}}}$ is originally defined. $\mathcal{H}_{ph}$ can be constructed as the projection of $\mathcal{H}$, via some suitable orthogonal projector operator P i.e., $\mathcal{H}_{ph}=P\mathcal{H}$. The restriction of H defined by (1.1) onto $\mathcal{H}_{ph}$ gives rise to the new Hamiltonian $H_{ph}=PHP$ (physical part of H) acting in $\mathcal{H}_{ph}$

$$\begin{aligned} H_{ph}f=\sum {E}_j \left\langle \varphi _j, f\right\rangle \,\varphi _j, \end{aligned}$$

(1.2)

where, the set of vectors ${{\mathcal {F}}}_\varphi =\left\{ \varphi _j=Pe_j\right\}$ loses the property of being an ONB of $\mathcal{H}_{ph}$ and, instead, it turns out to be a Parseval frame of $\mathcal{H}_{ph}$.

In this paper we explore further this situation, extending what we have found in [9] to unbounded operators, which are often more relevant in the analysis of concrete physical systems [14, 22].

The paper is divided in two parts: in Sect. 2 we describe our mathematical results, while Sect. 3 contains some detailed examples. More in details, after some preliminaries in Sect. 2.1, in Sect. 2.2 we introduce the Hamiltonian $H_{{\varphi }, \mathsf E}=\sum _{j\in {\mathbb {J}}}E_j \left\langle \varphi _j, \cdot \right\rangle \,\varphi _j$, for a PF $\left\{ \varphi _j\right\}$, and we study its domain, and we give conditions for $H_{{\varphi }, \mathsf E}$ to be, or not, bounded and self-adjoint. Its spectrum is then analyzed in Sect. 2.3.

In Sect. 3.1 we use a PF first introduced by Casazza and Christensen to define a particular $H_{{\varphi }, \mathsf E}$, and in particular to study its eigenvalues and eigenvectors. We also show how to introduce two different sets of ladder operators, $a_n$ and $V_n$. In particular, $a_n$ and its adjoint $a_n^*$ obey a truncated version of the canonical commutation relations which was first used, in our knowledge, in [11], and later extended in [3]. These truncated bosonic operators allow to interpolate between fermions and bosons, going from $n=2$ to $n\rightarrow \infty$. We will also comment that the operators $V_n$ may have an interesting role in signal analysis.

Section 3.2 contains a first part, where we propose a particular method to construct a PF out of an ONB, and then we apply this method to the Hamiltonian of a single electron in a strong magnetic field. Its eigenstates produce the so-called Landau levels, which have infinite degeneracy. This Hamiltonian is at the basis of the analysis of the quantum Hall effect. In particular, we will show that our approach could be seen as generating two different lattices somehow connected to these Landau levels.

2 Hamiltonians generated by Parseval frames

2.1 Parseval frames

Here all necessary facts about Parseval frames are presented in a form convenient for our exposition. More details can be found in [17, 24].

Let ${{\mathcal {K}}}$ be a complex infinite-dimensional separable Hilbert space with scalar product $\left\langle \cdot , \cdot \right\rangle$ linear in the second argument. Denote by ${\mathbb {J}}$ a generic countable index set such as ${\mathbb {Z}}$, ${\mathbb {N}}$, ${\mathbb {N}}\cup \{0\}$, etc.

A Parseval frame (PF for short) is a family of vectors ${{\mathcal {F}}}_\varphi =\left\{ \varphi _j, \,j\in {\mathbb {J}}\right\}$ in ${{\mathcal {K}}}$ which satisfies

$$\begin{aligned} \sum _{j\in {\mathbb {J}}}{\left| \left\langle \varphi _j, f\right\rangle \right| ^2}=||f||^2, \quad f\in {{\mathcal {K}}}. \end{aligned}$$

(2.1)

It follows from (2.1) that $\Vert \varphi _j\Vert \le {1}$ for $j\in {\mathbb {J}}$.

According to the Naimark dilation theorem, each PF in ${{\mathcal {K}}}$ can be extended to an orthonormal basis of a wider subspace $\mathcal{H}$.

Theorem 1

([23]) Let ${{\mathcal {F}}}_\varphi =\left\{ \varphi _j, \,j\in {\mathbb {J}}\right\}$ be a PF in a Hilbert space ${{\mathcal {K}}}$. Then there exists a complementary Hilbert space ${{\mathcal {M}}}$ and a PF ${{\mathcal {F}}}_\psi =\left\{ \psi _j, \,j\in {\mathbb {J}}\right\}$ in ${{\mathcal {M}}}$ such that

$$\begin{aligned} {{\mathcal {F}}}_h=\left\{ h_j=\varphi _j\oplus \psi _j, \ \ j\in {\mathbb {J}}\right\} \end{aligned}$$

(2.2)

is an ONB for $\mathcal{H}={{\mathcal {K}}}\oplus {{\mathcal {M}}}$.

It is also easy to prove, with a direct computation, that taken an ONB $\left\{ e_n\right\}$ of $\mathcal{H}$, and an orthogonal projector P, the set $\left\{ Pe_n\right\}$ is a PF in $\mathcal{H}_{ph}=P\mathcal{H}$, as already claimed in the Introduction.

The excess $e[{{\mathcal {F}}}_\varphi ]$ of a PF of ${{\mathcal {F}}}_\varphi =\left\{ \varphi _j, \,j\in {\mathbb {J}}\right\}$ is the greatest integer n such that n elements can be deleted from the frame ${{\mathcal {F}}}_\varphi$ and still leave a complete set, or $\infty$ if there is no upper bound to the number of elements that can be removed. It follows from [12, Lemma 4.1] and Theorem 1 that $e[{{\mathcal {F}}}_\varphi ]$ coincides with $\dim {{\mathcal {M}}}$, where ${{\mathcal {M}}}$ is the complementary Hilbert space in Theorem 1. The zero excess means that ${{\mathcal {F}}}_\varphi$ is an ONB of ${{\mathcal {K}}}$. The finite excess $e[{{\mathcal {F}}}_\varphi ]$ means that the index set ${\mathbb {J}}$ can be decomposed ${\mathbb {J}}={\mathbb {J}}_0\cup {\mathbb {J}}_1$ in such a way that ${{\mathcal {F}}}_\varphi ^0=\left\{ \varphi _j, j\in {\mathbb {J}}_0\right\}$ is a Riesz basis in ${{\mathcal {K}}}$ and ${\mathbb {J}}_1$ is a finite set [15].

Each PF ${{\mathcal {F}}}_\varphi$ determines an isometric operator $\theta _\varphi : {{\mathcal {K}}}\rightarrow \ell _2({\mathbb {J}})$:

$$\begin{aligned} \theta _\varphi {f}=\left\{ \left\langle \varphi _j, f\right\rangle \right\} _{j\in {\mathbb {J}}}, \qquad f\in {{\mathcal {K}}}, \end{aligned}$$

(2.3)

which is called an analysis operator associated with ${{\mathcal {F}}}_\varphi$.

The adjoint operator $\theta _\varphi ^*: \ell _2({\mathbb {J}}) \rightarrow {{\mathcal {K}}}$ of $\theta _\varphi$ is called a synthesis operator and it acts as follows

$$\begin{aligned} \theta _\varphi ^*\left\{ c_j\right\} =\sum _{j\in {\mathbb {J}}}c_j\varphi _j, \qquad \left\{ c_j\right\} \in \ell _2({\mathbb {J}}). \end{aligned}$$

(2.4)

Let $\theta _\varphi$ and $\theta _\psi$ be analysis operators associated with PF’s ${{\mathcal {F}}}_\varphi$ and ${{\mathcal {F}}}_\psi$ from Theorem 1. Then

$$\begin{aligned} \ell _2({\mathbb {J}})={{\mathcal {R}}}\left( \theta _\varphi \right) \oplus {{\mathcal {R}}}\left( \theta _\psi \right) , \end{aligned}$$

(2.5)

where ${{\mathcal {R}}}\left( \theta _\varphi \right)$ and ${{\mathcal {R}}}\left( \theta _\psi \right)$ are the image sets of the operators $\theta _\varphi$ and $\theta _\psi$, respectively.

By virtue of (2.4) and (2.5), for $\left\{ c_j\right\} \in \ell _2({\mathbb {J}})$, the following relation holds

$$\begin{aligned} \sum _{j\in {\mathbb {J}}}c_j\varphi _j=0 \quad \iff \quad \left\{ c_j\right\} \in \ker \theta _\varphi ^*=\ell _2({\mathbb {J}})\ominus {{\mathcal {R}}}\left( \theta _\varphi \right) ={{\mathcal {R}}}\left( \theta _\psi \right) . \end{aligned}$$

(2.6)

2.2 Operators $H_{{\varphi }\mathsf E}^{min}$ and $H_{{\varphi }\mathsf E}^{max}$

For a given PF ${{\mathcal {F}}}_{\varphi }=\left\{ \varphi _j, j\in {\mathbb {J}}\right\}$ and a sequence of real quantities $\textsf{E}=\left\{ E_j, j\in {\mathbb {J}}\right\}$, one can introduce a linear operator

$$\begin{aligned} H_{{\varphi }, \mathsf E}=\sum _{j\in {\mathbb {J}}}E_j \left\langle \varphi _j, \cdot \right\rangle \,\varphi _j \end{aligned}$$

(2.7)

in a Hilbert space ${{\mathcal {K}}}$. If $e[{{\mathcal {F}}}_{\varphi }]=0$ (i.e., ${{\mathcal {F}}}_{\varphi }$ is an ONB of ${{\mathcal {K}}}$), the quantities $E_j$ in (2.7) turns out to be eigenvalues of $H_{{\varphi }\mathsf E}$. If ${{\mathcal {F}}}_\varphi$ is a PF (but not an ONB), this fact does not hold in general (see Lemma 8 below).

The operator $H_{{\varphi }\mathsf E}$ may be unbounded and its domain of definition should be specified. There are two natural domains for $H_{{\varphi }\mathsf E}$ typically used in the literature:

$$\begin{aligned} \mathcal {D}_{min}= & {} \left\{ f\in {{\mathcal {K}}}\, \ \sum _{j\in {\mathbb {J}}}E_j^2 \left| \left\langle \varphi _j, f\right\rangle \right| ^2<\infty \right\} , \nonumber \\ \mathcal {D}_{max}= & {} \left\{ f\in {{\mathcal {K}}}\, \ \sum _{j\in {\mathbb {J}}}E_j\left\langle \varphi _j, f\right\rangle \,\varphi _j \quad \text{ converges } \text{ unconditionally } \text{ in } \quad {{\mathcal {K}}}\right\} .\qquad \end{aligned}$$

(2.8)

It follows from [24, Theorem 7.2 (b)] that

$$\begin{aligned} \mathcal {D}_{min}\subseteq \mathcal {D}_{max}. \end{aligned}$$

(2.9)

We observe that it may easily happen in (2.9) that $\mathcal {D}_{min}$ is indeed a proper subspaces of $\mathcal {D}_{max}$.

Example 2

Set ${\mathbb {J}}={\mathbb {Z}}\setminus \{0\}$ and consider the quantities $E_n=n^2$, $E_{-n}=0$ $(n\in {\mathbb {N}})$ and an ONB $\left\{ {e}_n\right\} _{n\in {\mathbb {N}}}$ of ${{\mathcal {K}}}$. In view of [24, Example 8.35], the set of vectors ${{\mathcal {F}}}_\varphi =\left\{ \varphi _j, \ j\in {\mathbb {J}}\right\}$, where

$$\begin{aligned} \varphi _n=\frac{1}{n}{e}_n, \qquad \varphi _{-n}=\sqrt{1-\frac{1}{n^2}}{e}_n \end{aligned}$$

is a PF. In this case, the series

$$\begin{aligned} \sum _{j\in {\mathbb {J}}}E_j \left\langle \varphi _j, f\right\rangle \,\varphi _j=\sum _{n\in {\mathbb {N}}}\left\langle {e}_n, f\right\rangle \, {e}_n \end{aligned}$$

converges unconditionally for all $f\in {{\mathcal {K}}}$. Hence, $\mathcal {D}_{max}={{\mathcal {K}}}$. On the other hand, $\mathcal {D}_{min}=\left\{ f\in {{\mathcal {K}}}: \sum _{n\in {\mathbb {N}}}n^2 \left| \left\langle {e}_n, f\right\rangle \right| ^2<\infty \right\}$ is a subset of ${{\mathcal {K}}}$. Therefore, $\mathcal {D}_{min}\subset \mathcal {D}_{max}$.

The specificity of frames gives rise to the following curious fact.

Proposition 3

For arbitrary unbounded set of quantities $\textsf{E}$ there are uncountably many PF’s ${{\mathcal {F}}}_\varphi$ such that $\mathcal {D}_{min}$ is trivial, i.e. $\mathcal {D}_{min}=\{0\}$.

Proof

Without loss of generality, one can assume that $\mathcal {H}=\ell ^2({\mathbb {J}})$, where ${\mathbb {J}}$ is the countable set of indices. Let $\mathcal {E}$ be an operator of multiplication by the set $\textsf{E}=\left\{ E_j, j\in {\mathbb {J}}\right\}$ in $\ell ^2({\mathbb {J}})$:

$$\begin{aligned} \mathcal {E}\left\{ c_j\right\} =\left\{ E_jc_j\right\} , \qquad \mathcal {D}(\mathcal {E})=\left\{ \left\{ c_j\right\} \in \ell ^2({\mathbb {J}}): \left\{ E_jc_j\right\} \in \ell ^2({\mathbb {J}}) \right\} . \end{aligned}$$

(2.10)

Since the set $\textsf{E}$ is unbounded (i.e. $\sup _{j\in {\mathbb {J}}}\left\{ \left| E_j\right| \right\} =\infty$) the self-adjoint operator $\mathcal {E}$ is unbounded in $\ell ^2({\mathbb {J}})$ and, by the extended version [1, Theorem 3.19] of the Schmüdgen theorem [31, Theorem 5.1], there are uncountably many infinite-dimensional subspaces ${{\mathcal {K}}}$ of $\ell ^2({\mathbb {J}})$ such that

$$\begin{aligned} \mathcal {D}(\mathcal {E})\cap {{\mathcal {K}}}=\{0\}. \end{aligned}$$

(2.11)

For given ${{\mathcal {K}}}$ satisfying (2.11) we denote by P the orthogonal projection operator in $\ell _2({\mathbb {J}})$ on ${{\mathcal {K}}}$ and consider the canonical ONB $\left\{ e_j, \ j\in {\mathbb {J}}\right\}$ of $\ell _2({\mathbb {J}})$. Then ${{\mathcal {F}}}_\varphi =\left\{ \varphi _j=Pe_j, \ j\in {\mathbb {J}}\right\}$ is a PF in $\mathcal {K}$. The associated analysis operator $\theta _\varphi$ (see (2.3)) maps ${{\mathcal {K}}}$ into $\ell ^2({\mathbb {J}})$ and

$$\begin{aligned} \theta _\varphi {f}=\left\{ \left\langle \varphi _j, f \right\rangle \right\} _{j\in {\mathbb {J}}}=\left\{ \left\langle e_j, f \right\rangle \right\} _{j\in {\mathbb {J}}}=\left\{ c_j\right\} =f, \qquad f=\left\{ c_j\right\} \in {{\mathcal {K}}}. \end{aligned}$$

Therefore, $\mathcal {R}\left( \theta _\varphi \right) ={{\mathcal {K}}}$ and, in view of (2.10), (2.11), $\left\{ E_jc_j\right\} \not \in \ell ^2({\mathbb {J}})$ for non-zero $f=\left\{ c_j\right\} \in \mathcal {K}$. This means that

$$\begin{aligned} \sum _{j\in {\mathbb {J}}}E_j^2 \left| \left\langle \varphi _j, f\right\rangle \right| ^2=\sum _{j\in {\mathbb {J}}}E_j^2 \left| \left\langle e_j, f\right\rangle \right| ^2=\sum _{j\in {\mathbb {J}}}E_j^2 |c_j|^2=\infty , \quad f\in {{\mathcal {K}}}, \ f\not =0 \end{aligned}$$

and $\mathcal {D}_{min}=\{0\}$. $\square$

Of course, the choice of PF’s ${{\mathcal {F}}}_\varphi$ in Proposition 3 should be very specific. Considering special classes of $\textsf{E}$ and ${{\mathcal {F}}}_\varphi$ one can guarantee that $\mathcal {D}_{min}$ coincides with $\mathcal {D}_{max}$ and is a dense set in ${{\mathcal {K}}}$. Few simple sufficient conditions are given below.

Proposition 4

The following assertions are true:

• (i) If $\sup _{j\in {\mathbb {J}}}\left\{ \left| E_j\right| \right\} <\infty$, then $\mathcal {D}_{min}=\mathcal {D}_{max}={{\mathcal {K}}}$;

• (ii) If the index set ${\mathbb {J}}$ of ${{\mathcal {F}}}_\varphi$ can be decomposed ${\mathbb {J}}={\mathbb {J}}_0\cup {\mathbb {J}}_1$ in such a way that $\left\{ \varphi _j, j\in {\mathbb {J}}_0\right\}$ is a Riesz basis in ${{\mathcal {K}}}$ and $\sup _{j\in {\mathbb {J}}_1}\left\{ \left| E_j\right| \right\} <\infty$, then $\mathcal {D}_{min}$ coincides with $\mathcal {D}_{max}$ and is a dense set in ${{\mathcal {K}}}$.

Proof

(i) Denote $\alpha =\sup _{j\in {\mathbb {J}}}\left\{ \left| E_j\right| \right\}$. Then

$$\begin{aligned} \sum _{j\in {\mathbb {J}}}E_j^2 \left| \left\langle \varphi _j, f\right\rangle \right| ^2\le \alpha ^2\sum _{j\in {\mathbb {J}}}\left| \left\langle \varphi _j, f\right\rangle \right| ^2=\alpha ^2\Vert f\Vert ^2, \qquad f\in {{\mathcal {K}}}. \end{aligned}$$

(2.12)

Therefore, $\mathcal {D}_{min}={{\mathcal {K}}}$. By virtue of (2.9), $\mathcal {D}_{min}=\mathcal {D}_{max}={{\mathcal {K}}}$.

(ii) Similarly to (2.12), $\sum _{j\in {\mathbb {J}}_1}E_j^2 \left| \left\langle \varphi _j, f\right\rangle \right| ^2\le \alpha ^2\sum _{j\in {\mathbb {J}}_1}\left| \left\langle \varphi _j, f\right\rangle \right| ^2\le \alpha ^2\Vert f\Vert ^2$, where $\alpha =\sup _{j\in {\mathbb {J}}_1}\left\{ \left| E_j\right| \right\}$ and $f\in {{\mathcal {K}}}.$ This means that the sets $\mathcal {D}_{min}$ and $\mathcal {D}_{max}$ defined by (2.8) do not change if we consider the smaller set ${\mathbb {J}}_0={\mathbb {J}}\setminus {\mathbb {J}}_1$ instead of ${\mathbb {J}}$.

Each Riesz basis $\left\{ \varphi _j, j\in {\mathbb {J}}_0\right\}$ is norm-bounded below (i.e., $\inf _{j\in {\mathbb {J}}}\Vert \varphi _j\Vert >0$). By [24, Theorem 8.36] this means that $\mathcal {D}_{min}=\mathcal {D}_{max}$. Let $\left\{ \psi _j, j\in {\mathbb {J}}_0\right\}$ be the biorthogonal Riesz basis for $\left\{ \varphi _j, j\in {\mathbb {J}}_0\right\}$. By virtue of (2.8), $\psi _j\in \mathcal {D}_{min}$ for all $j\in {\mathbb {J}}_0$. Hence, $\mathcal {D}_{min}$ is a dense set in ${{\mathcal {K}}}$. $\square$

In what follows we consider a slightly more general case assuming that $\mathcal {D}_{min}$ is dense in ${{\mathcal {K}}}$ and $\mathcal {D}_{min}\subseteq \mathcal {D}_{max}$, so that $\mathcal {D}_{max}$ is dense in ${{\mathcal {K}}}$ a fortiori.

Equipping (2.7) with domains $\mathcal {D}_{min}$ and $\mathcal {D}_{max}$ we define the following operators in ${{\mathcal {K}}}$:

$$\begin{aligned} \begin{array}{l} H_{{\varphi }\mathsf E}^{min}f=\sum _{j\in {\mathbb {J}}}E_j \left\langle \varphi _j, f\right\rangle \,\varphi _j, \qquad f\in \mathcal {D}\left( H_{{\varphi }\mathsf E}^{min}\right) =\mathcal {D}_{min}, \\ H_{{\varphi }\mathsf E}^{max}f=\sum _{j\in {\mathbb {J}}}E_j \left\langle \varphi _j, f\right\rangle \,\varphi _j, \qquad f\in \mathcal {D}\left( H_{{\varphi }\mathsf E}^{max}\right) =\mathcal {D}_{max}. \end{array} \end{aligned}$$

(2.13)

The operator $H_{{\varphi }\mathsf E}^{min}$ admits a simple interpretation with the use of Naimark dilation theorem. Namely, in the Hilbert space $\mathcal{H}={{\mathcal {K}}}\oplus {{\mathcal {M}}}$, we consider the ONB $\left\{ h_j, \ j\in {\mathbb {J}}\right\}$ defined by (2.2) and define a self-adjoint Hamiltonian

$$\begin{aligned} H_{h\textsf{E}}=\sum _{j\in {\mathbb {J}}}E_j \left\langle h_j, \cdot \right\rangle \,h_j, \qquad \mathcal {D}(H_{h\textsf{E}})=\left\{ f\in \mathcal{H}\, \ \sum _{j\in {\mathbb {J}}}E_j^2 \left| \left\langle h_j, f\right\rangle \right| ^2<\infty \right\} .\qquad \nonumber \\ \end{aligned}$$

(2.14)

By the construction, the set of eigenvalues of $H_{h\textsf{E}}$ coincides with $\textsf{E}=\{E_j\}$ and the corresponding eigenfunctions are $\{h_j\}$. Comparing (2.13) and (2.14) we arrive at the conclusion that

$$\begin{aligned} H_{{\varphi }\mathsf E}^{min}f=P_{{{\mathcal {K}}}}H_{h\textsf{E}}f, \qquad f\in \mathcal {D}\left( H_{{\varphi }\mathsf E}^{min}\right) =\mathcal {D}_{min}=\mathcal {D}\left( H_{h\textsf{E}}\right) \cap {{{\mathcal {K}}}}. \end{aligned}$$

(2.15)

Therefore, the operator $H_{{\varphi }\mathsf E}^{min}$ may be interpreted as the restriction of the Hamiltonian $H_{h\textsf{E}}$ acting in $\mathcal{H}={{\mathcal {K}}}\oplus {{\mathcal {M}}}$ to the physical space $\mathcal{H}_{ph}={{\mathcal {K}}}$, see (1.2).

It follows from (2.9) that

$$\begin{aligned} H_{{\varphi }\mathsf E}^{min}\subseteq {H_{{\varphi }\mathsf E}^{max}}. \end{aligned}$$

(2.16)

The densely defined operator $H_{{\varphi }\mathsf E}^{max}$ is symmetric in ${{\mathcal {K}}}$ since

$$\begin{aligned} \left\langle H_{{\varphi }\mathsf E}^{max}f, f\right\rangle =\sum _{j\in {\mathbb {J}}}E_j \left| \left\langle \varphi _j, f\right\rangle \right| ^2, \qquad f\in \mathcal {D}\left( H_{{\varphi }\mathsf E}^{max}\right) \end{aligned}$$

is real-valued. The same is true for $H_{{\varphi }\mathsf E}^{min}$ since (2.16) holds.

Corollary 5

If $H_{{\varphi }\mathsf E}^{min}$ is self-adjoint in ${{\mathcal {K}}}$, then $H_{{\varphi }\mathsf E}^{max}$ is also self-adjoint and $H_{{\varphi }\mathsf E}^{min}=H_{{\varphi }\mathsf E}^{max}$.

Proof

It follows immediately from the relation

$$\begin{aligned} H_{{\varphi }\mathsf E}^{min}\subseteq {H_{{\varphi }\mathsf E}^{max}}\subseteq \left( H_{{\varphi }\mathsf E}^{max}\right) ^*\subseteq \left( H_{{\varphi }\mathsf E}^{min}\right) ^*=H_{{\varphi }\mathsf E}^{min}. \end{aligned}$$

$\square$

Using Proposition 4, it is easy to derive sufficient conditions for self-adjointness of $H_{{\varphi }\mathsf E}^{min}$.

Corollary 6

The following assertions are true:

• (i) If $\sup _{j\in {\mathbb {J}}}\left\{ \left| E_j\right| \right\} <\infty$, then $H_{{\varphi }\mathsf E}^{min}$ is a bounded self-adjoint operator in ${{\mathcal {K}}}$;

• (ii) If $\sup _{j\in {\mathbb {J}}}\left\{ \left| E_j\right| \right\} =\infty$ and the index set ${\mathbb {J}}$ of ${{\mathcal {F}}}_\varphi$ can be decomposed ${\mathbb {J}}={\mathbb {J}}_0\cup {\mathbb {J}}_1$ in such a way that $\{\varphi _j, j\in {\mathbb {J}}_0\}$ is a Riesz basis in ${{\mathcal {K}}}$ and $\sup _{j\in {\mathbb {J}}_1}\left\{ \left| E_j\right| \right\} <\infty$, then $H_{{\varphi }\mathsf E}^{min}$ is an unbounded self-adjoint operator in ${{\mathcal {K}}}$.

Proof

(i). By Proposition 4 and (2.13), the symmetric operator $H_{{\varphi }\mathsf E}^{min}$ is defined on the whole space ${{\mathcal {K}}}$. Hence, $H_{{\varphi }\mathsf E}^{min}$ is a bounded self-adjoint operator.

(ii). By employing Proposition 4 once again, one gets that $H_{\varphi \mathsf E}^{min}$ is a densely defined operator in ${{\mathcal {K}}}$. For $f\in \mathcal {D}(H_{{\varphi }\mathsf E})$ we decompose

$$\begin{aligned} H_{{\varphi }\mathsf E}f=\sum _{j\in {\mathbb {J}}_0}E_j \left\langle \varphi _j, f\right\rangle \,\varphi _j+\sum _{j\in {\mathbb {J}}_1}E_j \left\langle \varphi _j, f\right\rangle \,\varphi _j, \end{aligned}$$

(2.17)

where ${{\mathcal {F}}}_\varphi ^0=\left\{ \varphi _j, \ j\in {\mathbb {J}}_0\right\}$ is a Riesz basis in ${{\mathcal {K}}}$ and $\sup _{j\in {\mathbb {J}}_1}\left\{ \left| E_j\right| \right\} <\infty$.

Denote by $S_0=\sum _{j\in {\mathbb {J}}_0}\left\langle \varphi _j, \cdot \right\rangle \,\varphi _j$ the frame operator of the Riesz basis ${{\mathcal {F}}}_\varphi ^0$. Then [17, Theorem 6.1.1]

$$\begin{aligned} \varphi _j=S_0^{1/2}e_j, \quad j\in {\mathbb {J}}_0, \end{aligned}$$

where $\left\{ e_j, \ j\in {\mathbb {J}}_0\right\}$ is an ONB of ${{\mathcal {K}}}$. Therefore, the first operator in (2.17),

$$\begin{aligned} A_0=\sum _{j\in {\mathbb {J}}_0}E_j \left\langle \varphi _j, \cdot \right\rangle \,\varphi _j, \quad \mathcal {D}(A_0)=\left\{ f\in {{\mathcal {K}}}\, \ \sum _{j\in {\mathbb {J}}_0}E_j^2 \left| \left\langle \varphi _j, f\right\rangle \right| ^2<\infty \right\} \end{aligned}$$

can be rewritten as follows

$$\begin{aligned} A_0=S_0^{1/2}H_{e\textsf{E}_0}S_0^{1/2}, \qquad H_{e\textsf{E}_0}=\sum _{j\in {\mathbb {J}}_0}E_j \left\langle e_j, \cdot \right\rangle \,e_j, \quad \textsf{E}_0=\left\{ E_j, j\in {\mathbb {J}}_0\right\} .\qquad \end{aligned}$$

(2.18)

By the construction, the operator $H_{e\textsf{E}_0}$ with the domain

$$\begin{aligned} \mathcal {D}(H_{e\textsf{E}_0})=\left\{ f\in {{\mathcal {K}}}\, \ \sum _{j\in {\mathbb {J}}_0}E_j^2 \left| \left\langle e_j, f\right\rangle \right| ^2<\infty \right\} \end{aligned}$$

is self-adjoint in ${{\mathcal {K}}}$. In view of (2.18), $\mathcal {D}(H_{e\textsf{E}_0})=S_0^{{1}/2}\mathcal {D}(A_0)$ and

$$\begin{aligned} \left\langle A_0f, g\right\rangle =\left\langle H_{e\textsf{E}_0}S_0^{1/2}f, S_0^{1/2}g\right\rangle , \qquad f,g\in \mathcal {D}(A_0). \end{aligned}$$

(2.19)

Relation (2.19) means that $A_0$ is self-adjoint. Moreover, as follows from the proof of Proposition 4, $\mathcal {D}(A_0)=\left\{ f\in {{\mathcal {K}}}\, \ \sum _{j\in {\mathbb {J}}_0}E_j^2 \left| \langle \varphi _j, f\rangle \right| ^2<\infty \right\} =\mathcal {D}_{min}=\mathcal {D}(H_{\varphi \textsf{E}}^{min})$.

On the other hand,

$$\begin{aligned} \sum _{j\in {\mathbb {J}}_1}E_j^2 \left| \left\langle \varphi _j, f\right\rangle \right| ^2\le \alpha ^2\sum _{j\in {\mathbb {J}}_1}\left| \left\langle \varphi _j, f\right\rangle \right| ^2<\alpha ^2\Vert f\Vert ^2, \quad f\in {{\mathcal {K}}}, \quad \alpha =\sup _{j\in {\mathbb {J}}_1}\left\{ \left| E_j\right| \right\} . \end{aligned}$$

This means (see, e.g., [24, Theorem 7.2]) that the second operator in (2.17)

$$\begin{aligned} A_1=\sum _{j\in {\mathbb {J}}_1}E_j \left\langle \varphi _j, \cdot \right\rangle \,\varphi _j \end{aligned}$$

is defined on ${{\mathcal {K}}}$ and it is symmetric (since $\left\langle A_1f, f \right\rangle =\sum _{j\in {\mathbb {J}}_1}E_j \left| \left\langle \varphi _j, f\right\rangle \right| ^2$ is real-valued for $f\in {{\mathcal {K}}}$). Hence, $A_1$ is a bounded self-adjoint operator in ${{\mathcal {K}}}$. This means that the operator $H_{{\varphi }\mathsf E}=A_0+A_1$ with the domain $\mathcal {D}(A_0)=\mathcal {D}\left( H_{{\varphi }\mathsf E}^{min}\right)$ is self-adjoint. $\square$

Remark 7

Similar results (in the case where ${{\mathcal {F}}}_{\varphi }$ involves a Riesz basis) can be obtained by the perturbation theory methods if the operator $A_1$ is sufficiently small with respect to the self-adjoint operator $A_0$ (see, e.g., [29, X.2]).

2.3 Spectrum of $H_{{\varphi }\mathsf E}$

In what follows, we suppose that $H_{{\varphi }\mathsf E}^{min}$ is a self-adjoint operator in ${{\mathcal {K}}}$. In this case, in view of Corollary 5, $H_{{\varphi }\mathsf E}:=H_{{\varphi }\mathsf E}^{min}=H_{{\varphi }\mathsf E}^{max}$ is a self-adjoint operator.

We consider $H_{{\varphi }\mathsf E}$ as a Hamiltonian generated by the pair $({{\mathcal {F}}}_\varphi , \textsf{E})$of a PF ${{\mathcal {F}}}_\varphi$and a set of real numbers $\textsf{E}$. As was mentioned above, the Hamiltonian $H_{{\varphi }\mathsf E}$ is the restriction to the physical space ${{\mathcal {K}}}$ of the Hamiltonian $H_{h\textsf{E}}$ (see (2.14)) acting in the wider Hilbert space $\mathcal{H}={{\mathcal {K}}}\oplus {{\mathcal {M}}}$. By virtue of (2.14), the point spectrum of $H_{h\textsf{E}}$ coincides with the set $\textsf{E}$ and $H_{h\textsf{E}}h_n=E_nh_n$$(n\in {\mathbb {J}})$. On the other hand:

Lemma 8

The following assertions are equivalent:

• (i) The relation $H_{{\varphi }\mathsf E}\varphi _n=E_n\varphi _n$ holds for some $E_n\in \textsf{E}$;

• (ii) The vectors of the PF ${{\mathcal {F}}}_\psi$ in Theorem 1 satisfy the following conditions:

  $$\begin{aligned} \sum _{j\in {\mathbb {J}}}E_j^2\left| \left\langle \psi _j, \psi _n\right\rangle \right| ^2<\infty , \qquad \sum _{j\in {\mathbb {J}}}E_j\left\langle \psi _j, \psi _n\right\rangle \varphi _j=0 \end{aligned}$$

  (2.20)

Proof

$(i)\rightarrow (ii)$. If $H_{{\varphi }\mathsf E}\varphi _n=E_n\varphi _n$, then $\varphi _n\in \mathcal {D}(H_{{\varphi }\mathsf E})\cap \mathcal {D}(H_{h\mathsf E})$ (see (2.15)). This means that the vector $\psi _n$ corresponding to $\varphi _n$ in (2.2) also belongs to $\mathcal {D}(H_{h\mathsf E})$. Therefore, $\sum _{j\in {\mathbb {J}}}E_j^2\left| \left\langle \psi _j, \psi _n\right\rangle \right| ^2=\sum _{j\in {\mathbb {J}}}E_j^2\left| \left\langle {h}_j, \psi _n\right\rangle \right| ^2<\infty .$ Further, the relation $H_{{h}\mathsf E}h_n=E_nh_n$ means that $P_{{{\mathcal {K}}}}H_{{h}\mathsf E}h_n=E_j\varphi _n$. Taking (2.2) and (2.14) into account, one gets

$$\begin{aligned} 0=P_{{{\mathcal {K}}}}H_{{h}\mathsf E}h_n-H_{{\varphi }\mathsf E}\varphi _n=\sum _{j\in {\mathbb {J}}}E_j\left\langle h_j, h_n\right\rangle \varphi _j-\sum _{j\in {\mathbb {J}}}E_j\left\langle \varphi _j, \varphi _n\right\rangle \varphi _j=\sum _{j\in {\mathbb {J}}}E_j\left\langle \psi _j, \psi _n\right\rangle \varphi _j\qquad \end{aligned}$$

(2.21)

that establishes (2.20).

$(ii)\rightarrow (i)$. The first part in (2.20) and (2.14) mean that $\psi _n\in \mathcal {D}(H_{h\mathsf E})$. Hence, $\varphi _n=h_n-\psi _n$ belongs to $\mathcal {D}(H_{{\varphi }\mathsf E})\cap \mathcal {D}(H_{h\mathsf E})$. Reasoning similarly to (2.21), we obtain

$$\begin{aligned} 0=\sum _{j\in {\mathbb {J}}}E_j\left\langle \psi _j, \psi _n\right\rangle \varphi _j=P_{{{\mathcal {K}}}}H_{{h}\mathsf E}h_n-H_{{\varphi }\mathsf E}\varphi _n=E_nP_{{{\mathcal {K}}}}h_n-H_{{\varphi }\mathsf E}\varphi _n=E_n\varphi _n-H_{{\varphi }\mathsf E}\varphi _n \end{aligned}$$

that completes the proof. $\square$

If $\Vert \varphi _n\Vert =1$, then $\psi _n=0$ and the conditions (2.20) are clearly satisfied. In such case, $E_n$ turns out to be an eigenvalue of $H_{{\varphi }\mathsf E}$.

Lemma 8 shows that the set $\mathsf E$ not always coincides with the point spectrum of $H_{{\varphi }\mathsf E}$. For this reason, we will say that $\textsf{E}$ is a set of quasi-eigenvalues of $H_{\varphi \textsf{E}}$.

Since $H_{{\varphi }\mathsf E}$ is assumed to be self-adjoint, one may believe that $H_{{\varphi }\mathsf E}$ can also be presented in a form similar to (2.14):

$$\begin{aligned} H_{{\varphi }\mathsf E}=\sum _{j\in {\mathbb {J}}}\,E_j' \left\langle {e}_j', \cdot \right\rangle {e}_j', \qquad \mathcal {D}(H_{{\varphi }\mathsf E})=\left\{ f\in {{\mathcal {K}}}\, \ \sum _{j\in {\mathbb {J}}}E_j'^2\left| \left\langle {e}_j', f\right\rangle \right| ^2<\infty \right\} ,\qquad \nonumber \\ \end{aligned}$$

(2.22)

where ${{\mathcal {F}}}_{e'}=\left\{ {e}_j'\in {{\mathcal {K}}}, \ j\in {{\mathbb {J}}}\right\}$ is an ONB of ${{\mathcal {K}}}$ and $\mathsf{E'}=\left\{ {E}_j', \ j\in {{\mathbb {J}}}\right\}$ is a set of real numbers. The formula (2.22) is more convenient for spectral analysis because it immediately gives the set $\mathsf{E'}$ of eigenvalues of $H_{{\varphi }\mathsf E}$. This means, according what proposed in [9] for bounded operators, that $({{\mathcal {F}}}_\varphi ,\textsf{E})$ is $\textsf{E}-$connected to ${{\mathcal {F}}}_{e'}$. Indeed, under the assumptions here, $H_{{\varphi }\mathsf E}=\sum _{j\in {\mathbb {J}}}E_j\left\langle \varphi _j, \cdot \right\rangle \varphi _j=\sum _{j\in {\mathbb {J}}}\,E_j' \left\langle {e}_j', \cdot \right\rangle {e}_j'$.

We recall before formulating the next statement that the PF ${{\mathcal {F}}}_\varphi$ can be extended to an ONB ${{\mathcal {F}}}_h$ in $\mathcal{H}={{\mathcal {K}}}\oplus {{\mathcal {M}}}$ by adding a complementary PF ${{\mathcal {F}}}_\psi$ of ${{\mathcal {M}}}$ (as in Theorem 1).

Theorem 9

Let $H_{{\varphi }\mathsf E}$ be a Hamiltonian generated by the pair $({{\mathcal {F}}}_\varphi , \textsf{E})$, where $\textsf{E}=\{E_j, j\in {\mathbb {J}}\}$ is a strictly increasing sequence $\ldots<E_j<{E_{j+1}}\ldots$ and let the operator

$$\begin{aligned} B=\sum _{j\in {\mathbb {J}}}E_j \left\langle \varphi _j, \cdot \right\rangle \,\psi _j \, \ {{\mathcal {K}}}\rightarrow {{\mathcal {M}}}\end{aligned}$$

(2.23)

be bounded. Then $H_{{\varphi }\mathsf E}$ has the form (2.22) and its discrete spectrum $\sigma _{disc}(H_{{\varphi }\mathsf E})$ coincides with $\mathsf{E'}$.

Proof

We can assume, without loss of generality, that ${\mathbb {J}}={\mathbb {N}}$. Since $\textsf{E}=\left\{ E_j, j\in {\mathbb {N}}\right\}$ is strictly increasing, there exists $\lambda =\lim _{j\rightarrow \infty }E_j$. When $\lambda$ is less than infinity, $\textsf{E}$ is a bounded set, and the condition of boundedness of B in (2.23) is automatically fulfilled. Moreover the self-adjoint operator

$$\begin{aligned} \lambda {I}-H_{h\textsf{E}}=\sum _{j\in {\mathbb {N}}}\left( \lambda -E_j\right) \left\langle h_j, \cdot \right\rangle \,h_j, \end{aligned}$$

where $H_{h\textsf{E}}$ is defined by (2.14), is positive, bounded and compact¹ in $\mathcal{H}$. The same properties hold true for the operator $\lambda {I}-H_{\varphi \textsf{E}}=P_{{{\mathcal {K}}}}\left( \lambda {I}-H_{h\textsf{E}}\right) P_{{\mathcal {K}}}$ acting in ${{\mathcal {K}}}$. Therefore, there exists an ONB $\left\{ {e}_j'\right\} _{j\in {\mathbb {N}}}$ of ${{\mathcal {K}}}$ formed by eigenvectors ${e}_j'$:

$$\begin{aligned} \left( \lambda {I}-H_{\varphi \textsf{E}}\right) {e}_j'=\mu _j{e}_j', \qquad j\in {\mathbb {N}}, \end{aligned}$$

(2.24)

corresponding to the decreasing sequence of eigenvalues $\mu _j$. The set $\left\{ \mu _j\right\} _{j\in {\mathbb {N}}}$ constitutes the discrete spectrum of $\lambda {I}-H_{\varphi \textsf{E}}$. It follows from (2.24) that $H_{\varphi \textsf{E}}{e}_j'=(\lambda -\mu _j){e}_j'$. Therefore, the bounded operator $H_{\varphi \textsf{E}}$ can be defined by (2.22) with $E_j'=\lambda -\mu _j$ and its discrete spectrum $\sigma _{disc}(H_{{\varphi }\mathsf E})$ coincides with $\mathsf{E'}=\left\{ E_j'=\lambda -\mu _j, \ j\in {\mathbb {N}}\right\}$.

Assume now that $\lambda =\lim _{j\rightarrow \infty }E_j=\infty$. Then the operators $H_{h\textsf{E}}$ and $H_{{\varphi }\mathsf E}$ are both semi-bounded from below, and they are interconnected through the relation (2.15). The operator $H_{h\textsf{E}}$ has a compact resolvent (since its eigenvectors $\left\{ h_j\right\}$ form an ONB of $\mathcal{H}$ and the corresponding eigenvalues $\left\{ E_j\right\}$ are an increasing sequence tending to $\infty$). In this case, by means of [30, Theorem XIII.64], the set

$$\begin{aligned} Y_{b{H_{h\textsf{E}}}}=\left\{ f\in \mathcal {D}(H_{h\textsf{E}}): \Vert f\Vert \le {1}, \ \Vert H_{h\textsf{E}}f\Vert \le {b}\right\} , \end{aligned}$$

is compact in $\mathcal {H}$ for every $b\ge {0}$. Consider the similar set associated with $H_{\varphi \textsf{E}}$

$$\begin{aligned} Y_{b{H_{\varphi \textsf{E}}}}=\left\{ f\in \mathcal {D}(H_{\varphi \textsf{E}}): \Vert f\Vert \le {1}, \ \Vert H_{\varphi \textsf{E}}f\Vert \le {b}\right\} . \end{aligned}$$

In view of (2.15) and (2.23), for all $f\in {Y_{b{H_{\varphi \textsf{E}}}}},$

$$\begin{aligned} P_{{{\mathcal {M}}}}H_{h\textsf{E}}f=P_{{{\mathcal {M}}}}\sum _{j\in {\mathbb {N}}}E_j \left\langle h_j, f \right\rangle \,h_j=\sum _{j\in {\mathbb {N}}}E_j \left\langle \varphi _j, f \right\rangle \,\psi _j=Bf, \end{aligned}$$

where $P_{{{\mathcal {M}}}}$ is the orthogonal projection operator on ${{\mathcal {M}}}$ in $\mathcal{H}$. Furthermore,

$$\begin{aligned} \Vert H_{h\textsf{E}}f\Vert ^2=\Vert P_{{{\mathcal {K}}}}H_{h\textsf{E}}f\Vert ^2+\Vert P_{{{\mathcal {M}}}}H_{h\textsf{E}}f\Vert ^2=\Vert H_{\varphi \textsf{E}}f\Vert ^2+\Vert Bf\Vert ^2\le {b^2+c^2}, \quad f\in {Y_{b{H_{\varphi \textsf{E}}}}},\qquad \nonumber \\ \end{aligned}$$

(2.25)

where $c=\sup _{f\in {Y_{b{H_{\varphi \textsf{E}}}}}}\Vert Bf\Vert \le \Vert B\Vert <\infty$. Therefore,

$$\begin{aligned} Y_{b{H_{\varphi \textsf{E}}}}\subset {Y_{\sqrt{b^2+c^2}{H_{h\textsf{E}}}}}. \end{aligned}$$

(2.26)

Consider a convergent sequence $\{f_n\}$ in ${{\mathcal {K}}}$, where $f_n{\in }Y_{b{H_{\varphi \textsf{E}}}}$ and denote $f=\lim {f_n}$. Obviously, $f\in {{\mathcal {K}}}$ and $\Vert f\Vert \le {1}$. Moreover, $f\in {Y_{\sqrt{b^2+c^2}{H_{h\textsf{E}}}}}$ by virtue of (2.26) and the compactness of ${Y_{\sqrt{b^2+c^2}{H_{h\textsf{E}}}}}$. This means that $f\in \mathcal {D}(H_{h\textsf{E}})\cap {{\mathcal {K}}}=\mathcal {D}(H_{\varphi \textsf{E}})$. Assume that $\Vert H_{\varphi \textsf{E}}f\Vert >b$, i.e. $\Vert H_{\varphi \textsf{E}}f\Vert ^2=b^2+\varepsilon$ ($\varepsilon >0$). By virtue of (2.25), $\Vert Bf\Vert ^2\le {c^2-\varepsilon }$ that contradicts to the definition of c. Hence, $\Vert H_{\varphi \textsf{E}}f\Vert \le {b}$ and $f\in {Y_{b{H_{\varphi \textsf{E}}}}}$. We verify that $Y_{b{H_{\varphi \textsf{E}}}}$ is a closed set in ${{\mathcal {K}}}$. This fact, the compactness of $Y_{b{H_{h\textsf{E}}}}$, and (2.26) mean that $Y_{b{H_{\varphi \textsf{E}}}}$ is a compact set in ${{\mathcal {K}}}$ for each $b\ge {0}$. Applying again [30, Theorem XIII.64] we obtain that $H_{\varphi \textsf{E}}$ has a compact resolvent in ${{\mathcal {K}}}$. This means that formula (2.22) holds, the spectrum of $H_{\varphi \textsf{E}}$ is discrete² and it coincides with $\textsf{E}'$. $\square$

Remark 10

The statement in Theorem 9 can be readily extended to cover the scenario of an increasing sequence $\ldots {\le }E_j{\le }{E_{j+1}}\le \ldots$ provided that eigenvalues have finite multiplicities.

Theorem 9 establishes a sufficient condition for the existence of the discrete spectrum of $H_{\varphi \textsf{E}}$ without the need for explicit construction. The Min-Max principle [32, p. 265] can be utilized to compute the eigenvalues. Additionally, an alternative method is presented below, which is specifically tailored to the properties of operators $H_{\varphi \textsf{E}}$.

Proposition 11

Let $H_{{\varphi }\mathsf E}$ be a Hamiltonian generated by the pair $({{\mathcal {F}}}_\varphi , \textsf{E})$ and let ${{\mathcal {R}}}\left( \theta _\varphi \right)$ be the image set of the analysis operator $\theta _\varphi$ associated with ${{\mathcal {F}}}_\varphi$ (see (2.3)). Then $\mu \in \sigma _p(H_{{\varphi }\mathsf E})$ if and only if there exists a sequence $\left\{ c_j\right\} \in \mathcal {R}\left( \theta _\varphi \right)$ such that

$$\begin{aligned} \left\{ \left( E_j-\mu \right) c_j\right\} \in \ell _2({\mathbb {J}})\ominus {{\mathcal {R}}}\left( \theta _\varphi \right) =\mathcal {R}\left( \theta _\psi \right) . \end{aligned}$$

(2.27)

The corresponding eigenvector of $H_{{\varphi }\mathsf E}$ coincides with $f=\sum _{j\in {\mathbb {J}}}c_j \varphi _j$.

Proof

Assume that $H_{{\varphi }\mathsf E}f=\mu {f}$ for some $f\in \mathcal {D}(H_{{\varphi }\mathsf E})$. Since ${{\mathcal {F}}}_\varphi$ is a PF, $f=\sum _{j\in {\mathbb {J}}}\left\langle \varphi _j, f \right\rangle \varphi _j$ and the relation $H_{{\varphi }\mathsf E}f=\mu {f}$ takes the form

$$\begin{aligned} \sum _{j\in {\mathbb {J}}}\left( E_j-\mu \right) \left\langle \varphi _j, f \right\rangle \varphi _j=\sum _{j\in {\mathbb {J}}}\left( E_j-\mu \right) c_j\varphi _j=0, \qquad c_j=\left\langle \varphi _j, f \right\rangle . \end{aligned}$$

(2.28)

In view of (2.3), $\theta _\varphi {f}=\left\{ c_j\right\}$, i.e., the sequence $\left\{ c_j\right\}$ belongs to $\mathcal {R}\left( \theta _\varphi \right)$. Moreover, $\left\{ E_jc_j\right\} \in \ell _2({\mathbb {J}})$ since $f\in \mathcal {D}(H_{{\varphi }\mathsf E})$. Combining (2.6) with (2.28) we obtain (2.27).

Conversely, if (2.27) holds, then $\left\{ \left( E_j-\mu \right) c_j\right\} \in \ell _2({\mathbb {J}})$ and $\sum _{j\in {\mathbb {J}}}\left( E_j-\mu \right) c_j\varphi _j=0$, by virtue of (2.6). The sequence $\left\{ c_j\right\} \in \mathcal {R}\left( \theta _\varphi \right)$ determines a vector $f=\sum _{j\in {\mathbb {J}}}c_j \varphi _j\in {{\mathcal {K}}}$, where $c_j=\left\langle \varphi _j, f \right\rangle$. It follows from (2.27) that $\{E_j\left\langle \varphi _j, f\right\rangle \}\in \ell _2({\mathbb {J}})$. This means that $f\in \mathcal {D}(H_{{\varphi }\mathsf E})$ and the relation $\sum _{j\in {\mathbb {J}}}\left( E_j-\mu \right) c_j\varphi _j=\sum _{j\in {\mathbb {J}}}\left( E_j-\mu \right) \left\langle \varphi _j, f\right\rangle \varphi _j=0$ is equivalent to $H_{{\varphi }\mathsf E}f-\mu {f}=0$. $\square$

Typically, a PF ${{\mathcal {F}}}_\varphi =\left\{ \varphi _j, j\in {\mathbb {J}}\right\}$ comprises a Riesz basis component ${{\mathcal {F}}}_\varphi ^0=\left\{ \varphi _j, j\in {\mathbb {J}}_0\right\}$ (${\mathbb {J}}_0\subset {\mathbb {J}}$). This is especially true when ${{\mathcal {F}}}_\varphi$ has finite excess. In this case, the Hamiltonian $H_{{\varphi }\mathsf E}$ generated by the pair $({{\mathcal {F}}}_\varphi , \textsf{E})$ can be decomposed:

$$\begin{aligned} H_{{\varphi }\mathsf E}=A_0+A_1, \qquad A_i=\sum _{j\in {\mathbb {J}}_i}E_j\left\langle \varphi _j, \cdot \right\rangle \varphi _j. \end{aligned}$$

(2.29)

An additional analysis leads to the following:

Proposition 12

Assume that a PF ${{\mathcal {F}}}_\varphi =\left\{ \varphi _j, j\in {\mathbb {J}}\right\}$ can be decomposed ${{\mathcal {F}}}_\varphi ={{\mathcal {F}}}_\varphi ^0\cup {{\mathcal {F}}}_\varphi ^1$ in such a way that ${{\mathcal {F}}}_\varphi ^0=\left\{ \varphi _j, j\in {\mathbb {J}}_0\right\}$ and ${{\mathcal {F}}}_\varphi ^1=\left\{ \varphi _j, j\in {\mathbb {J}}_1\right\}$ are, respectively, a Riesz basis and a frame sequence in ${{\mathcal {K}}}$. Denote by $S_0$ the frame operator for ${{\mathcal {F}}}_\varphi ^0$ and suppose that $\sup _{j\in {\mathbb {J}}_1}\left\{ \left| E_j\right| \right\} <\infty$. Then, for all $f\in \mathcal {D}(H_{{\varphi }\mathsf E})$,

$$\begin{aligned} H_{{\varphi }\mathsf E}f=S_0^{1/2}H_{e\textsf{E}_0}S_0^{1/2}f+(I-S_0)^{1/2}H_{e\textsf{E}_1}(I-S_0)^{1/2}f \end{aligned}$$

(2.30)

where the self-adjoint operator $H_{e\textsf{E}_0}$ is defined by (2.18), the set $\{e_j, j\in {\mathbb {J}}_1\}$ is a PF of the subspace ${{\mathcal {K}}}_1=\overline{\textsf{span}\ {{\mathcal {F}}}_\varphi ^1}=\mathcal {R}(I-S_0)$, and $H_{e\textsf{E}_1}=\sum _{j\in {\mathbb {J}}_1}E_j \left\langle e_j, \cdot \right\rangle \,e_j$ is a bounded self-adjoint operator in ${{\mathcal {K}}}_1$.

Proof

It follows from the proof of Corollary 6 that the operator $A_0$ in (2.29) coincides with $S_0^{1/2}H_{e\textsf{E}_0}S_0^{1/2}$, where $H_{e\textsf{E}0}$ is defined by (2.18).

The operator $I-S_0$ is nonnegative in ${{\mathcal {K}}}$ since

$$\begin{aligned} \left( \left( I-S_0\right) f, f\right) =\Vert f\Vert ^2-\sum _{j\in {\mathbb {J}}_0}\left| \left\langle \varphi _j, f \right\rangle \right| ^2=\sum _{j\in {\mathbb {J}}_1}\left| \left\langle \varphi _j, f \right\rangle \right| ^2\ge {0}, \quad f\in {{\mathcal {K}}}.\qquad \end{aligned}$$

(2.31)

This means that the square root $\left( I-S_0\right) ^{1/2}$ exists. Further, the relation (2.31) implies that

$$\begin{aligned} \ker (I-S_0)={{\mathcal {K}}}\ominus {{\mathcal {K}}}_1, \qquad {{\mathcal {K}}}_1=\overline{\textsf{span}\ {{\mathcal {F}}}_\varphi ^1}. \end{aligned}$$

Hence, ${{\mathcal {K}}}_1$ coincides with $\overline{\mathcal {R}(I-S_0)}$ and it is a reducing subspace for $I-S_0$. Denote by $(I-S_0)|_{{{\mathcal {K}}}_1}$ the restriction of $I-S_0$ onto ${{\mathcal {K}}}_1$. The relation

$$\begin{aligned} (I-S_0)f=\sum _{j\in {\mathbb {J}}}\left\langle \varphi _j, f \right\rangle \varphi _j -\sum _{j\in {\mathbb {J}}_0}\left\langle \varphi _j, f\right\rangle \varphi _j=\sum _{j\in {\mathbb {J}}_1}\left\langle \varphi _j, f\right\rangle \varphi _j, \quad f\in {{\mathcal {K}}}_1 \end{aligned}$$

implies that $(I-S_0)|_{{{\mathcal {K}}}_1}$ is a frame operator of the frame ${{\mathcal {F}}}_\varphi ^1$ in the Hilbert space ${{\mathcal {K}}}_1$. Hence, the inverse operator $\left( (I-S_0)_{{{\mathcal {K}}}_1}\right) ^{-1}: {{\mathcal {K}}}_1 \rightarrow {{\mathcal {K}}}_1$ is bounded and $\overline{\mathcal {R}(I-S_0)}= {\mathcal {R}(I-S_0)}$.

It follows from [17, Theorem 6.1.1] that the elements $\varphi _j\in {{\mathcal {F}}}_\varphi ^1$ have the form

$$\begin{aligned} \varphi _j=(I-S_0)^{1/2}e_j, \qquad j\in {\mathbb {J}}_1, \end{aligned}$$

where $\{e_j, j\in {\mathbb {J}}_1\}$ is a PF of ${{\mathcal {K}}}_1$. Moreover, repeating the proof of the relation (2.18) for the case of operator $A_1$ acting in ${{\mathcal {K}}}_1$ we get $A_1=(I-S_0)^{1/2}H_{e\textsf{E}_1}(I-S_0)^{1/2}$. Here, the operator $H_{e\textsf{E}_1}$ is bounded self-adjoint in ${{\mathcal {K}}}_1$ due to the part (i) of Corollary 6. $\square$

Considering $H_{{\varphi }\mathsf E}$ as a perturbation of $A_0$ by $A_1$ and supposing that $A_1$ is sufficiently small with respect to $A_0$ one can expect the coincidence of essential spectra of $H_{{\varphi }\mathsf E}$ and $A_0$. For example, If ${{\mathcal {F}}}_\varphi$ has a finite excess, then $I-S_0$ is a compact operator, and therefore the second operator in (2.30) is also compact. In such a case, the classical Weyl theorem [32, p. 182] implies that $\sigma _{ess}(H_{{\varphi }\mathsf E})=\sigma _{ess}\left( A_0+A_1\right) =\sigma _{ess}(A_0)$.

3 Examples

This section is devoted to a detailed analysis of two examples of our previous results, with some preliminary applications to quantum mechanics.

3.1 Hamiltonians generated by the Casazza-Christensen frame

In general, a PF with infinite excess may not contain a Riesz basis as a subset. If a subsequence of the frame elements is allowed to converge to 0 in norm, then it is easy to construct a frame that does not contain a Riesz basis. However, answering a similar question for frames that are norm-bounded below is much more complicated. An example of such a PF that is norm-bounded below, but does not contain a Schauder basis, was first constructed by Casazza and Christensen [15]. Our aim now is to investigate Hamiltonians generated by that PF and discuss their possible physical applications.

Let ${{\mathcal {K}}}_n$ ($n\in {\mathbb {N}}$) be a n-dimensional Hilbert space with ONB $\left\{ {e}_1^{(n)}, {e}_2^{(n)}, \ldots {e}_n^{(n)}\right\}$. Then the set ${{\mathcal {F}}}_\varphi ^{(n)}=\left\{ \varphi _j^{(n)}, j\in {\mathbb {J}}_n\right\}$ where ${\mathbb {J}}_n=\{1, 2, \ldots , n, n+1\}$ and

$$\begin{aligned} \varphi _j^{(n)}={e}_j^{(n)}-\frac{1}{n}\sum _{i=1}^{n}{e}_i^{(n)}, \quad 1\le {j}\le {n}, \qquad \varphi _{n+1}^{(n)}=\frac{1}{\sqrt{n}}\sum _{i=1}^{n}{e}_i^{(n)}, \end{aligned}$$

(3.1)

is a PF for ${{\mathcal {K}}}_n$ [17, Lemma 7.5.1]. The operator

$$\begin{aligned} {H_{{\varphi _n}\textsf{E}_n}}=\sum _{j\in {\mathbb {J}}_n}E_j^{(n)} \left\langle \varphi _j^{(n)}, \cdot \right\rangle \,\varphi _j^{(n)}=\sum _{j=1}^{n+1}E_j^{(n)} \left\langle \varphi _j^{(n)}, \cdot \right\rangle \,\varphi _j^{(n)}, \qquad \textsf{E}_n=\left\{ E_j^{(n)}, j\in {\mathbb {J}}_n\right\} . \end{aligned}$$

(3.2)

is a bounded self-adjoint operator in ${{\mathcal {K}}}_n.$

For the PF ${{\mathcal {F}}}_\varphi ^{(n)}$, the complementary Hilbert space ${{\mathcal {M}}}_n$ can be chosen as ${\mathbb {C}}$ and the complementary PF ${{\mathcal {F}}}_\psi ^{(n)}=\left\{ \psi _j^{(n)}, j\in {\mathbb {J}}_n\right\}$ in (2.2) has the form [25]:

$$\begin{aligned} \psi _j^{(n)}=\frac{1}{\sqrt{n}}, \quad j=1,\ldots {n}, \qquad \psi _{n+1}^{(n)}=0. \end{aligned}$$

(3.3)

Consider the direct sum ${{\mathcal {K}}}=\left( \sum _{n=1}^\infty \oplus {{\mathcal {K}}}_n\right) _{\ell _2}.$ The Hilbert space ${{\mathcal {K}}}$ consists of sequences $f=(f_1, f_2, \ldots )$ for which³$f_n\in {{\mathcal {K}}}_n$ and $\sum _{n=1}^\infty \Vert f_n\Vert _{{{\mathcal {K}}}_n}^2<\infty$. The scalar product in ${{\mathcal {K}}}$ is defined as follows

$$\begin{aligned} \left\langle f, g\right\rangle =\sum _{n=1}^\infty \left\langle f_n, g_n \right\rangle _{{{\mathcal {K}}}_n}, \qquad f, g \in {{\mathcal {K}}}. \end{aligned}$$

Since ${{\mathcal {K}}}=\left( \sum _{n=1}^\infty \oplus {{\mathcal {K}}}_n\right) _{\ell _2}$, the union of PF’s ${{\mathcal {F}}}_{\varphi }=\bigcup _{n=1}^\infty {{\mathcal {F}}}_{\varphi }^{(n)}$ is a PF for ${{\mathcal {K}}}$ [17, Theorem 7.5.2]. The complementary Hilbert space ${{\mathcal {M}}}$ in Theorem 1 for the PF ${{\mathcal {F}}}_\varphi$ can be chosen as the direct sum

$$\begin{aligned} {{\mathcal {M}}}=\left( \sum _{n=1}^\infty \oplus {{\mathcal {M}}}_n\right) _{\ell _2}=\left( \sum _{n=1}^\infty \oplus {\mathbb {C}}\right) _{\ell _2}=\ell _2({\mathbb {N}}). \end{aligned}$$

The PF ${{\mathcal {F}}}_\psi$ in ${{\mathcal {M}}}$ coincides with the union of PF’s: ${{\mathcal {F}}}_{\psi }=\bigcup _{n=1}^\infty {{\mathcal {F}}}_{\psi }^{(n)}$.

Denote

$$\begin{aligned} \textsf{E}=\bigcup _{n=1}^\infty \textsf{E}_n=\left\{ E_j^{(n)}, \ n\in {\mathbb {N}}, \ 1\le {j}\le {n+1}\right\} . \end{aligned}$$

For the PF ${{\mathcal {F}}}_\varphi$ and $\textsf{E}$ we consider the operator $H_{{\varphi }\mathsf E}^{min}: {{\mathcal {K}}}\rightarrow {{\mathcal {K}}}$ defined by (2.13).

$$\begin{aligned}{} & {} H_{{\varphi }\mathsf E}^{min}=\sum _{n=1}^\infty \sum _{j=1}^{n+1}E_j^{(n)}\left\langle \varphi _j^{(n)}, \cdot \right\rangle \,\varphi _j^{(n)}, \\{} & {} \mathcal {D}\left( H_{{\varphi }\mathsf E}^{min}\right) =\left\{ f\in {{\mathcal {K}}}\, \ \sum _{n=1}^\infty \sum _{j=1}^{n+1}\left( E_j^{(n)}\right) ^2 \left| \left\langle \varphi _j^{(n)}, f\right\rangle \right| ^2<\infty \right\} . \end{aligned}$$

Taking into account the decomposition ${{\mathcal {K}}}=\left( \sum _{n=1}^\infty \oplus {{\mathcal {K}}}_n\right) _{\ell _2}$ and (3.2) one can rewrite the last formulas as follows

$$\begin{aligned} H_{{\varphi }\mathsf E}^{min}=\sum _{n=1}^\infty \oplus {H_{{\varphi _n}\textsf{E}_n}}, \qquad \mathcal {D}\left( {H_{{\varphi _n}\textsf{E}_n}^{min}}\right) =\left\{ f\in {{\mathcal {K}}}\, \ \sum _{n=1}^\infty \Vert H_{\varphi _n E_n}f_n\Vert _{{{\mathcal {K}}}_n}^2<\infty \right\} . \end{aligned}$$

The obtained formula means that $H_{{\varphi }\mathsf E}^{min}$ is self-adjoint in ${{\mathcal {K}}}$ (since $H_{{\varphi _n}\textsf{E}_n}$ are bounded self-adjoint operators in ${{\mathcal {K}}}_n$ for all $n\in {\mathbb {N}}$). By Corollary 5, $H_{{\varphi }\mathsf E}:=H_{{\varphi }\mathsf E}^{min}=H_{{\varphi }\mathsf E}^{max}$ is a Hamiltonian generated by the pair $({{\mathcal {F}}}_\varphi , \textsf{E})$.

As the PF ${{\mathcal {F}}}_\varphi$ does not contain a Riesz basis [15], Proposition 12 cannot be applied for the investigation of $H_{{\varphi }\mathsf E}$. Nonetheless, one can attempt to use Theorem 9 and Proposition 11.

Assume that $\left\{ E_j^{(n)}\right\}$ is a strictly increasing sequence, i.e.

$$\begin{aligned} E_1^{(1)}<E_2^{(1)}<E_1^{(2)}<E_{2}^{(2)}<E_3^{(2)}\ldots<E_1^{(n)}<E_2^{(n)}\ldots<E_n^{(n)}<E_{n+1}^{(n)}<\ldots , \end{aligned}$$

where $E_j^{(n)} \rightarrow \infty$. Taking (3.3) into account one gets that the operator $B: {{\mathcal {K}}}\rightarrow \ell _2({\mathbb {N}})$ defined by (2.23) has the form

$$\begin{aligned} Bf=\left\{ \frac{1}{\sqrt{n}}\sum _{j=1}^{n}{E_j^{(n)}} \left\langle \varphi _j^{(n)}, f\right\rangle \right\} _{n=1}^\infty , \qquad f\in \mathcal {D}(B)\subseteq {{\mathcal {K}}}. \end{aligned}$$

(3.4)

If

$$\begin{aligned} \sup _{n\in {\mathbb {N}}, \ 1\le {j}\le {n}}\frac{1}{\sqrt{n}}E_j^{(n)}<\infty , \end{aligned}$$

then the right-hand side of (3.4) belongs to $\ell _2({\mathbb {N}})$ for every $f\in {{\mathcal {K}}}$ and the operator B is bounded. In this case, applying Theorem 9 we arrive at the conclusion that $H_{{\varphi }\mathsf E}$ has a discrete spectrum. Explicit calculation of the discrete spectrum can be carried out using Proposition 11. Building on the argumentation presented in [9], we conclude that the discrete spectrum of $H_{\varphi \textsf{E}}$ consists of the original quantities⁴$\left\{ E_{n+1}^{(n)}\right\} _{n=1}^\infty$ and the union of the solutions $\mu _1, \ldots \mu _{n-1}$, $n\ge 2$, of the equations

$$\begin{aligned} \frac{1}{E_1^{(n)}-\mu }+\frac{1}{E_2^{(n)}-\mu }+\ldots +\frac{1}{E_{n}^{(n)}-\mu }=0, \qquad n\ge {2}. \end{aligned}$$

The corresponding eigenfunctions are

$$\begin{aligned} \varphi _{n+1}^{(n)}=\frac{1}{n}\sum _{i=1}^{n}{e_i^{(n)}} \quad \bigg (\text{ for } \text{ the } \text{ eigenvalues } \quad E_{n+1}^{(n)}\bigg ) \quad \text{ and } \quad f_j=\sum _{i=1}^{n}\frac{1}{E_i^{(n)}-\mu _j}{e}_i^{(n)}, \end{aligned}$$

for the eigenvalues $\mu _j$, $1\le {j}\le {n-1}$ ($n\ge {2}$).

This example can be used to produce a natural settings in the realm of signal analysis. For that, we need first to introduce some ladder operators. In particular, we will now introduce the horizontal ladder operators $a_n$ and $a_n^*$, acting on ${{\mathcal {K}}}_n$, and the vertical ladder operators $V_n$ and $V_n^*$, mapping ${{\mathcal {K}}}_{n+1}$ into ${{\mathcal {K}}}_{n}$ and vice-versa.

First of all we define

$$\begin{aligned} a_ne_j^{(n)}=\left\{ \begin{array}{ll} 0, \qquad \,\,\,&{} \text{ if } j=1 \\ \sqrt{j-1}\,e^{(n)}_{j-1}, \qquad &{} \text{ if } \, j=2,3,\ldots ,n, \end{array}\right. \end{aligned}$$

whose adjoint is

$$\begin{aligned} a_n^* e_j^{(n)}=\left\{ \begin{array}{ll} \sqrt{j}\,e^{(n)}_{j+1}, \quad &{} \quad \text{ if } \, j=1,2,\ldots ,n-1,\\ 0, \quad &{}\quad \text{ if } \; j=n. \end{array}\right. \end{aligned}$$

These operators, already introduced in [7] in connection with a biological system, satisfy the following commutation rule:

$$\begin{aligned} \left[ a_n,a_n^*\right] =I_{{{\mathcal {K}}}_n}-nP_n^{(n)}, \end{aligned}$$

where $P_n^{(n)}f=\left\langle e_n^{(n)},f\right\rangle _{{{\mathcal {K}}}_n}e_n^{(n)}$, for all $f\in {{\mathcal {K}}}_n$, and where $I_{{{\mathcal {K}}}_n}$ is the identity operator on ${{\mathcal {K}}}_n$.

The action of $a_n$ on the vectors $\varphi _j^{(n)}$ defined by (3.1) is as follows:

$$\begin{aligned} a_n\varphi _j^{(n)}=\left\{ \begin{array}{ll} -{\tilde{e}}^{(n)},&{} \quad \,\,\, \text{ if } j=1 \\ \sqrt{j-1}\,\varphi ^{(n)}_{j-1}+ \frac{\sqrt{j-1}}{\sqrt{n}}\,\varphi _{n+1}^{(n)} -{\tilde{e}}^{(n)}, &{}\quad \,\,\, \text{ if } j=2,3,\ldots ,n,\\ \sqrt{n}\, {\tilde{e}}^{(n)}, &{}\quad \,\,\, \text{ if } j=n+1, \end{array}\right. \end{aligned}$$

where

$$\begin{aligned} {\tilde{e}}^{(n)}=\frac{1}{n} \sum _{i=1}^{n-1} \sqrt{i}\,{e}_i^{(n)}. \end{aligned}$$

It is easy (but not so relevant) to deduce also the action of $a_n^*$ on $\varphi _j^{(n)}$.

We can further introduce the operator $V_{n+1}:{{\mathcal {K}}}_{n+1}\rightarrow {{\mathcal {K}}}_n$ as follows:

$$\begin{aligned} V_{n+1}f=\sum _{j=1}^{n+1}\left\langle e_j^{(n+1)},f\right\rangle _{{{\mathcal {K}}}_{n+1}}\,\varphi _j^{(n)}, \qquad f\in {{\mathcal {K}}}_{n+1}. \end{aligned}$$

In particular, we see that $V_{n+1}e_j^{(n+1)}=\varphi _j^{(n)}$. The adjoint of $V_{n+1}$ can be easily deduced. It is an operator mapping ${{\mathcal {K}}}_n$ into ${{\mathcal {K}}}_{n+1}$ as follows:

$$\begin{aligned} V_{n+1}^*g=\sum _{j=1}^{n+1}\left\langle \varphi _j^{(n)},g\right\rangle _{{{\mathcal {K}}}_{n}}\,e_j^{(n+1)}, \qquad g\in {{\mathcal {K}}}_n. \end{aligned}$$

In this case, it is clear that $V_{n+1}^*\varphi _j^{(n)}\ne e_j^{(n+1)}$, in general. However, due to the fact that $\big \{\varphi _j^{(n)}\big \}_{j=1}^{n+1}$ is a PF in ${{\mathcal {K}}}_n$, it is possible to check that $V_{n+1}V_{n+1}^*=I_{{{\mathcal {K}}}_n}$, while $V_{n+1}^*V_{n+1}\ne {I}_{{{\mathcal {K}}}_{n+1}}$.

As for the possible interpretation of this example, and the ladder-like operators $a_n$, $V_{n+1}$ and their adjoints, a natural look at this framework is in terms of signal analysis: ${{\mathcal {K}}}_n$ is the set of signals with n bits. If a signal f is $f=e_1^{(n)}$, this means that only the first bit is “on”, in a signal of n bits. Analogously, if $f=\alpha _1e_1^{(n)}+\alpha _ne_n^{(n)}$, then the first and the last bits are “on”, with two weights $\alpha _1$ and $\alpha _n$, with $\left| \alpha _1\right| ^2+\left| \alpha _n\right| ^2=1$. The operators $a_n$ and $a_n^*$ switch on and off the various bits of a signal with n bits, while $V_{n+1}$ and $V_{n+1}^*$ add or remove bits from the signal. Also, going from $\big \{e_j^{(e)}\big \}$ to $\big \{\varphi _j^{(e)}\big \}$ in this context is rather natural: a frame is what is often used in signal analysis to take into account possible loss of information during the transmission of the signal. The Hamiltonian can be seen as the energy of the signal, with various contributions arising from different possible lengths of the signals.

3.2 Relations with regular pseudo-bosons

This section is focused on a class of examples of PFs connected to the so-called regular pseudo-bosons [2, 4], which are suitable deformations of the bosonic ladder operators $c=\frac{1}{\sqrt{2}}\left( x+\frac{d}{dx}\right)$ and $c^*=\frac{1}{\sqrt{2}}\left( x-\frac{d}{dx}\right)$.

The starting point here is a bounded operator X with bounded inverse $X^{-1}$ and an ONB ${{\mathcal {F}}}_e=\left\{ e_n\in {{\mathcal {K}}}, \, n\in {\mathbb {J}}={\mathbb {N}}\cup \{0\}\right\}$ of a Hilbert space ${{\mathcal {K}}}$. Then we have

Proposition 13

If $\Vert X^*X\Vert <1$, then the sets

$$\begin{aligned} {{\mathcal {F}}}_{\varphi }=\left\{ \varphi _n=X^*e_n, \,n\in {\mathbb {J}}\right\} , \qquad {{\mathcal {F}}}_{{{\tilde{\varphi }}}}=\left\{ {{\tilde{\varphi }}}_n=\left( I-X^*X\right) ^{1/2}e_n, \,n\in {\mathbb {J}}\right\} \end{aligned}$$

are Riesz bases of ${{\mathcal {K}}}$, with dual bases

$$\begin{aligned} {{\mathcal {F}}}_{\psi }=\left\{ \psi _n=X^{-1}e_n, \,n\in {\mathbb {J}}\right\} , \qquad {{\mathcal {F}}}_{{{\tilde{\psi }}}}=\left\{ {{\tilde{\psi }}}_n=\left( I-X^*X\right) ^{-\,1/2}e_n, \,n\in {\mathbb {J}}\right\} . \end{aligned}$$

Moreover, the set ${{\mathcal {F}}}_{ex}^\varphi ={{\mathcal {F}}}_{\varphi }\cup {{\mathcal {F}}}_{{{\tilde{\varphi }}}}$ is a PF of ${{\mathcal {K}}}$.

If $\Vert X^*X\Vert >1$ then the set ${{\mathcal {F}}}_{\widetilde{{{\widetilde{\psi }}}}}=\left\{ \widetilde{{{\widetilde{\psi }}}_n}=\left( I-\left( X^*X\right) ^{-1}\right) ^{1/2}e_n, \,n\in {\mathbb {J}}\right\}$ is a Riesz basis with dual

$$\begin{aligned} {{\mathcal {F}}}_{\widetilde{{{\widetilde{\varphi }}}}}=\left\{ \widetilde{{{\widetilde{\varphi }}}_n}=\left( I-\left( X^*X\right) ^{-1}\right) ^{-\,1/2}e_n, \,n\in {\mathbb {J}}\right\} \end{aligned}$$

and the set ${{\mathcal {F}}}_{ex}^\psi ={{\mathcal {F}}}_{\psi }\cup {{\mathcal {F}}}_{\widetilde{{{\widetilde{\psi }}}}}$ is a PF of ${{\mathcal {K}}}$.

Proof

If $\left\| X^*X\right\| <1$, then $I-X^*X$ is a bounded positive operator on ${{\mathcal {K}}}$, and there exist a positive square root $\left( I-X^*X\right) ^{1/2}$ and its inverse $\left( I-X^*X\right) ^{-1/2}$ that are bounded operators on ${{\mathcal {K}}}$. Hence the Riesz basis nature of the pairs ${{\mathcal {F}}}_{\varphi }$, ${{\mathcal {F}}}_{\psi }$ and ${{\mathcal {F}}}_{{{\tilde{\varphi }}}}$, ${{\mathcal {F}}}_{{{\tilde{\psi }}}}$ is clear. To prove that ${{\mathcal {F}}}_{ex}^\varphi$ is a PF, let us put $U=X^*X$, which is bounded with bounded inverse and self adjoint, and let us write a generic $f\in {{\mathcal {K}}}$ as $f=Uf+(I-U)f$. We have

$$\begin{aligned} \left\langle Uf,f\right\rangle =\left\langle Xf,Xf\right\rangle =\sum _{n\in {\mathbb {J}}}\left| \left\langle Xf,e_n\right\rangle \right| ^2 =\sum _{n\in {\mathbb {J}}}\left| \left\langle f,\varphi _n\right\rangle \right| ^2. \end{aligned}$$

Moreover, since $I-U$ is a positive operator,

$$\begin{aligned} (I-U)f= & {} (I-U)^{1/2}\left( (I-U)^{1/2}f\right) =(I-U)^{1/2}\sum _{n\in {\mathbb {J}}}\left\langle e_n,(I-U)^{1/2}f\right\rangle e_n\\= & {} \sum _{n\in {\mathbb {J}}}\left\langle {{\tilde{\varphi }}}_n,f\right\rangle {{\tilde{\varphi }}}_n, \end{aligned}$$

so that $\left\langle (I-U)f,f\right\rangle =\sum _{n\in {\mathbb {J}}}\left| \left\langle {{\tilde{\varphi }}}_n, f\right\rangle \right| ^2$. Hence

$$\begin{aligned} \Vert f\Vert ^2=\left\langle Uf,f\right\rangle +\left\langle (I-U)f,f\right\rangle =\sum _{n\in {\mathbb {J}}}\left| \left\langle f,\varphi _n\right\rangle \right| ^2+\sum _{n\in {\mathbb {J}}}\left| \left\langle {{\tilde{\varphi }}}_n,f\right\rangle \right| ^2, \end{aligned}$$

which implies our claim: ${{\mathcal {F}}}_{ex}^\varphi$ is a PF.

The proof of the second part of the proposition is similar to this, and will not be repeated.

$\square$

Remark

It is clear that if the sets ${{\mathcal {F}}}_{{{\tilde{\varphi }}}}$ and ${{\mathcal {F}}}_{{{\tilde{\psi }}}}$ are well defined, then the other sets, ${{\mathcal {F}}}_{\widetilde{{{\widetilde{\varphi }}}}}$ and ${{\mathcal {F}}}_{\widetilde{{{\widetilde{\psi }}}}}$, are not. The point is that only one between $I-U$ and $I-U^{-1}$ can be positive. This does not exclude the possibility to extend the above construction to consider both these possibilities. But we will not investigate further this point here.

3.2.1 An explicit construction

As it is discussed in [2, 4], in particular, pseudo-bosons can be seen as suitable deformations of the ladder operators c and $c^*$ we introduced before. It is well known that these operators can be used to diagonalize the Hamiltonian of a quantum harmonic oscillator $H_0=\frac{1}{2}\left( p^2+x^2\right)$, where p and x are respectively the momentum and the position operators (both self-adjoint). In fact, after some algebra we can write $H_0=c^*c+\frac{1}{2}\,I$, and its eigenstates $e_n$ can be constructed by fixing first the vacuum $e_0$, i.e. a vector in ${{\mathcal {K}}}$ such that $ce_0=0$, and then acting on it with powers of $c^*$: $e_n=\frac{\left( c^*\right) ^n}{\sqrt{n!}}\,e_0$. Then, ${{\mathcal {F}}}_e=\left\{ e_n, \,n\ge 0\right\}$ is an ONB of ${{\mathcal {K}}}$. In particular, in the position representation, where c and $c^*$ are the differential operators already introduced, we have ${{\mathcal {K}}}={{{\mathcal {L}}}^2({\mathbb {R}})}$ and

$$\begin{aligned} e_n(x)=\frac{1}{\sqrt{2^nn!\sqrt{\pi }}}H_n(x)\,e^{-\frac{x^2}{2}}, \end{aligned}$$

where $H_n(x)$ is the n-th Hermite polynomial.

Remark

Despite the apparent simplicity of the system (the well known harmonic oscillator), it is maybe useful to stress that this is not necessarily trivial. Indeed, we could think of $H_0$ as the single electron Hamiltonian of a two-dimensional gas of electrons in a strong magnetic field, orthogonal to a fixed plane, when expressed in suitable variables. This is the physical system which is behind the Landau levels, and the fractional quantum Hall effect, see [16, 27] for instance, and has attracted a lot of interest in the past years, [10, 20, 26, 28] among the others. In most of these papers, the possibility of modifying a single wave function in the so-called lowest Landau level (LLL) is used to construct the wave function for the gas of electrons, localized at different lattice sites and minimizing the energy of the gas.

In what we will do now, we are inspired by the possibility of magnetically translating a wave function of the LLL while staying in the same energetic level. However, rather than using only translation operators, we will consider a combination of multiplication and translation operators. More explicitly, let K and T be the operators defined as follows:

$$\begin{aligned} Kf(x)=m(x)f(x), \qquad Tf(x)=f(x-\alpha ), \qquad f\in {{{\mathcal {L}}}^2({\mathbb {R}})}. \end{aligned}$$

Here $\alpha >0$, fixed, and m(x) is a complex-valued, smooth⁵ function satisfying $0<m\le |m(x)|\le M<1$ in ${\mathbb {R}}$. These operators are bounded, with bounded inverse. In particular, T is unitary:

$$\begin{aligned} K^{-1}f(x)=\frac{1}{m(x)}\,f(x), \qquad K^*f(x)=\overline{m(x)}\,f(x), \qquad T^*f(x)=T^{-1}f(x)=f(x+\alpha ). \end{aligned}$$

Now, if we put $X^*=TK$, we get the following results:

$$\begin{aligned} X^*f(x)=m(x-\alpha )\,f(x-\alpha ), \qquad Xf(x)=\overline{m(x)}f(x+\alpha ), \end{aligned}$$

together with

$$\begin{aligned} (X^{-1})^*f(x)=\frac{1}{m(x)}\,f(x+\alpha ), \qquad X^{-1}f(x)=\frac{1}{\overline{m(x-\alpha )}}\,f(x-\alpha ). \end{aligned}$$

The operators $X^*X$ and $XX^*$ turn out to be both multiplication operators:

$$\begin{aligned} X^*Xf(x)=|m(x-\alpha )|^2f(x), \qquad XX^*f(x)=|m(x)|^2f(x). \end{aligned}$$

Because of our assumption on m, we have $|m(x-\alpha )|^2\le {M^2}<1$, and therefore $\Vert X^*X\Vert <1$: we are in the first case of Proposition 13, so that the sets ${{\mathcal {F}}}_{\widetilde{{{\widetilde{\varphi }}}}}$ and ${{\mathcal {F}}}_{\widetilde{{{\widetilde{\psi }}}}}$ cannot be defined. Still we find

$$\begin{aligned} \varphi _n(x)=m(x-\alpha )e_n(x-\alpha ), \qquad \psi _n(x)=\frac{1}{\overline{m(x-\alpha )}}e_n(x-\alpha ) \end{aligned}$$

and

$$\begin{aligned} {{\tilde{\varphi }}}_n(x)=\sqrt{1-|m(x-\alpha )|^2}\,e_n(x), \qquad {{\tilde{\psi }}}_n(x)=\frac{1}{\sqrt{1-|m(x-\alpha )|^2}}\,e_n(x). \end{aligned}$$

Biorthonormality (in pairs) of these functions is manifest, while the fact that ${{\mathcal {F}}}_{ex}^\varphi ={{\mathcal {F}}}_{\varphi }\cup {{\mathcal {F}}}_{{{\tilde{\varphi }}}}$ is a PF is not as clear, but it is a consequence of Proposition 13.

Following [4] it is easy to find the ladder operators for ${{\mathcal {F}}}_\varphi$ and ${{\mathcal {F}}}_{{{\tilde{\varphi }}}}$, and for their dual Riesz bases. We introduce the operators $a_\varphi$, $b_\varphi$, $a_{{{\tilde{\varphi }}}}$ and $b_{{{\tilde{\varphi }}}}$ as follows:

$$\begin{aligned} a_\varphi f(x)=X^*c(X^*)^{-1}f(x), \qquad b_\varphi f(x)=X^*c^*(X^*)^{-1}f(x), \end{aligned}$$

and

$$\begin{aligned}{} & {} a_{{{\tilde{\varphi }}}} f(x)=\left( I-X^*X\right) ^{1/2}c\left( I-X^*X\right) ^{-1/2}f(x), \\ {}{} & {} b_{{{\tilde{\varphi }}}} f(x)=\left( I-X^*X\right) ^{1/2}c^*\left( I-X^*X\right) ^{-1/2}f(x), \end{aligned}$$

$\forall f(x)\in {{{\mathcal {S}}}({\mathbb {R}})}$, the set of the $C^\infty$, fast decreasing, functions (the Schwartz space). Simple computations allow us to deduce the following expressions:

$$\begin{aligned} a_\varphi =c-\frac{1}{\sqrt{2}}\left( \alpha +\frac{m'(x-\alpha )}{m(x-\alpha )}\right) , \qquad b_\varphi =c^*-\frac{1}{\sqrt{2}}\left( \alpha -\frac{m'(x-\alpha )}{m(x-\alpha )}\right) , \end{aligned}$$

while

$$\begin{aligned} a_{{{\tilde{\varphi }}}}=c-\frac{1}{\sqrt{2}}\frac{q'(x)}{q(x)}, \qquad b_{{{\tilde{\varphi }}}}=c^*+\frac{1}{\sqrt{2}}\frac{q'(x)}{q(x)}, \end{aligned}$$

where $q(x)=\sqrt{1-|m(x-\alpha )|^2}$. Incidentally we observe that we can rewrite $\frac{q'(x)}{q(x)}=\frac{d(\log (q(x)))}{dx}$.

Pseudo-bosonic operators are useful since they act as ladder operators on the families of function deduced before. In particular, we have

$$\begin{aligned} a_\varphi \varphi _n(x)=\sqrt{n}\varphi _{n-1}(x), \qquad b_\varphi \varphi _n(x)=\sqrt{n+1}\varphi _{n+1}(x), \end{aligned}$$

and similarly

$$\begin{aligned} a_{{{\tilde{\varphi }}}}{{\tilde{\varphi }}}_n(x)=\sqrt{n}{{\tilde{\varphi }}}_{n-1}(x), \qquad b_{{{\tilde{\varphi }}}}{{\tilde{\varphi }}}_n(x)=\sqrt{n+1}{{\tilde{\varphi }}}_{n+1}(x), \end{aligned}$$

with the understanding that $\varphi _{-1}(x)={{\tilde{\varphi }}}_{-1}(x)=0$. As a consequence, the various $\varphi _n(x)$ are eigenstates of $N_\varphi =b_\varphi a_\varphi$, while each ${{\tilde{\varphi }}}_n(x)$ is an eigenstate of $N_{{{\tilde{\varphi }}}}=b_{{{\tilde{\varphi }}}} a_{{{\tilde{\varphi }}}}$, both with eigenvalue n. If we further compute the adjoints of these operators, then we obtain ladder operators for the dual families, ${{\mathcal {F}}}_\psi$ and ${{\mathcal {F}}}_{{{\tilde{\psi }}}}$, [2, 4].

Defining now the vectors $\Phi _n$ as

$$\begin{aligned} \Phi _n(x)=\left\{ \begin{array}{c} \varphi _n(x), \qquad \,\,\,\,\,\, n=0,1,2,3,\ldots \\ {\tilde{\varphi }}_{-n-1}(x), \qquad \,\,\, {n=-1,- 2, - 3,\ldots } \end{array}\right. \end{aligned}$$

and the set ${{\mathcal {F}}}_\Phi =\left\{ \Phi _n(x),\,n\in {\mathbb {Z}}\right\} {={{\mathcal {F}}}_\varphi \cup {{\mathcal {F}}}_{{\tilde{\varphi }}}}$, it is easy to check that, taken any $f(x)\in {{{\mathcal {L}}}^2({\mathbb {R}})}$,

$$\begin{aligned} \sum _{n\in {\mathbb {Z}}}\left\langle \Phi _n,f\right\rangle \Phi _n(x)= & {} \sum _{n=0}^\infty \langle \varphi _n,f\rangle \varphi _n(x)+\sum _{n=0}^\infty \left\langle {\tilde{\varphi }}_n,f\right\rangle {\tilde{\varphi }}_n(x)\\ {}= & {} X^*Xf(x)+(1-X^*X)f(x)=f(x), \end{aligned}$$

as it should (since ${{\mathcal {F}}}_\varphi \cup {{\mathcal {F}}}_{{\tilde{\varphi }}}$ is a PF in ${{\mathcal {K}}}={{{\mathcal {L}}}^2({\mathbb {R}})}$). Now, given a set of (real) numbers $E_n$, $n\in {\mathbb {Z}}$, we can consider an operator

$$\begin{aligned} H=\sum _{n\in {\mathbb {Z}}}E_n\left\langle \Phi _n,\cdot \right\rangle \Phi _n(x)=H_1+H_2, \end{aligned}$$

where

$$\begin{aligned} H_1=\sum _{n=0}^\infty E_n\left\langle \varphi _n,\cdot \right\rangle \varphi _n(x), \qquad H_2=\sum _{n=0}^\infty E_{-(n+1)}\left\langle {{\tilde{\varphi }}}_n,\cdot \right\rangle {{\tilde{\varphi }}}_n(x). \end{aligned}$$

If we now fix $E_n=E_{-(n+1)}=n$, then

$$\begin{aligned} H_1=\sum _{n=0}^\infty n\left\langle \varphi _n,\cdot \right\rangle \varphi _n(x)=N_\varphi \sum _{n=0}^\infty \left\langle \varphi _n,\cdot \right\rangle \varphi _n(x), \end{aligned}$$

while $H_2=\sum _{n=0}^\infty n\left\langle {{\tilde{\varphi }}}_n,\cdot \right\rangle {{\tilde{\varphi }}}_n(x)=N_{{{\tilde{\varphi }}}} \sum _{n=0}^\infty \left\langle {{\tilde{\varphi }}}_n,\cdot \right\rangle {{\tilde{\varphi }}}_n(x)$.

As for a possible interpretation of these two terms, we can go back to our previous remark, and to the explicit expressions of the functions $\varphi _n(x)$ and ${{\tilde{\varphi }}}_n(x)$. In particular, while these latter are proportional (via the weight function $\sqrt{1-|m(x-\alpha )|^2}$) to the $e_n(x)$, the functions $\varphi _n(x)$ are again proportional (but via the other weight function $m(x-\alpha )$) to the translated version of the $e_n(x)$. Then, while $H_2$ can be seen as the single-electron deformed Hamiltonian for a particle in a sort of modified lowest Landau level, $H_1$ can be seen as its shifted version (with a different weight function). This can be interesting in connection with the crystals constructed out of the single electron, as in [10, 20, 26, 28], since it could produce two different lattices, one shifted with respect to the other. In condensed matter, lattices of this kind are useful, like the so-called reciprocal lattices.

These examples do not cover all possible applications of the general strategy proposed in this paper. More results, and more applications, are part of our future plans.