Inverse spectral problems for Dirac-type operators with global delay on a star graph

Abstract

We introduce Dirac-type operators with a global constant delay on a star graph consisting of m equal edges. For our introduced operators, we formulate an inverse spectral problem that is recovering the potentials from the spectra of two boundary value problems on the graph with a common set of boundary conditions at all boundary vertices except for a specific boundary vertex $v_{0}$ (called the root). For simplicity, we restrict ourselves to the constant delay not less than the edge length of the graph. Under the assumption that the common boundary conditions are of the Robin type and they are known and pairwise linearly independent, the uniqueness theorem is proven and a constructive procedure for solving the proposed inverse problem is obtained.

A. Appendix

Definition A.1. Let $b>0$. An entire function $S(\lambda )$ of exponential type is called a sine-type function of exponential type b, if there exist positive constants $c_{1}$, $c_{2}$ and K such that

$$\begin{aligned} c_{1}\exp (b|\text {Im}\lambda |)\le |S(\lambda )|\le c_{2}\exp (b|\text {Im}\lambda |),\quad |\text {Im}\lambda |>K. \end{aligned}$$

Lemma A.1

Let $H_{j}\!\in \!\mathbb {C}$ and $H_{j}\ne \pm i$. Then the function $v_{1,j}(\lambda )\!=\!\cos \lambda -H_{j}\sin \lambda$ is a sine-type function of exponential type 1.

Proof

Obviously, the following inequality holds:

$$\begin{aligned} |v_{1,j}(\lambda )|\le (1+|H_{j}|)\exp (|\text {Im}\lambda |),\quad \forall \lambda \in \mathbb {C}. \end{aligned}$$

(A.1)

Using Euler’s formula, one can get

$$\begin{aligned} v_{1,j}(\lambda )=\frac{i(H_{j}-i)}{2}\exp (i\lambda )-\frac{i(H_{j}+i)}{2}\exp (-i\lambda ). \end{aligned}$$

(A.2)

Since

$$\begin{aligned} \lim _{\text {Im}\lambda \rightarrow +\infty }\left| \frac{H_{j}-i}{H_{j}+i}\exp (2i\lambda )\right| =0, \end{aligned}$$

there exist a positive constant $K_{1}$ such that

$$\begin{aligned} \left| \frac{H_{j}-i}{H_{j}+i}\exp (2i\lambda )-1\right|>\frac{1}{2}, \quad \text {Im}\lambda >K_{1}. \end{aligned}$$

(A.3)

According to (A.2) and (A.3), we have

$$\begin{aligned} |v_{1,j}(\lambda )|\ge \frac{|H_{j}+i|}{4}\exp (\text {Im}\lambda ), \quad \text {Im}\lambda >K_{1}. \end{aligned}$$

(A.4)

Similarly, there exist a positive constant $K_{2}$ such that

$$\begin{aligned} |v_{1,j}(\lambda )|\ge \frac{|H_{j}-i|}{4}\exp (-\text {Im}\lambda ), \quad \text {Im}\lambda <-K_{2}. \end{aligned}$$

(A.5)

Let $K=\max \{K_{1},\;K_{2}\}$ and $c_{1}=\min \left\{ \frac{|H_{j}+i|}{4},\;\frac{|H_{j}-i|}{4}\right\}$. Then, according to (A.4) and (A.5), we have

$$\begin{aligned} |v_{1,j}(\lambda )|\ge c_{1}\exp (|\text {Im}\lambda |), \quad |\text {Im}\lambda |>K. \end{aligned}$$

(A.6)

Combining (A.1) and (A.6), we arrive at the assertion. $\square$

Put

$$\begin{aligned} f_{1}(\lambda )=\sin \lambda \prod _{j=2}^{m}v_{1,j}(\lambda ),\quad f_{2}(\lambda )=-\cos \lambda \prod _{j=2}^{m}v_{1,j}(\lambda ). \end{aligned}$$

(3.7)

From the relations (2.11), (2.14) and (2.24), it follows that

$$\begin{aligned} f_{1}'(\lambda )=\Delta _{1}^{0}(\lambda ),\quad f_{2}'(\lambda )=\Delta _{2}^{0}(\lambda ). \end{aligned}$$

(3.8)

Note that the following property of sine-type function is established in [20] (see also Introduction in [23]):

If f is a sine-type function of exponential type b, then uniformly in $x\in \mathbb {R}$ the following limit exist

$$\begin{aligned} \lim _{y\rightarrow +\infty }\frac{f'(x+iy)}{f(x+iy)}=-ib,\quad \lim _{y\rightarrow -\infty }\frac{f'(x+iy)}{f(x+iy)}=ib. \end{aligned}$$

According to the property, we arrive at the following assertion.

Lemma A.2

If the function $f(\lambda )$ is a sine-type function of exponential type b, then the derivative $f'(\lambda )$ is also a sine-type function of exponential type b.

Proof

By the property above, we have

$$\begin{aligned} \lim _{|\text {Im}\lambda |\rightarrow \infty }\left| \frac{f'(\lambda )}{f(\lambda )}\right| =b. \end{aligned}$$

Thus, there exist a positive constant K such that

$$\begin{aligned} \frac{b}{2}|f(\lambda )|<|f'(\lambda )|<2b|f(\lambda )|,\quad |\text {Im}\lambda |>K. \end{aligned}$$

Since $f(\lambda )$ is a sine-type function of exponential type b along with the last inequality, we arrive at the assertion. $\square$

Using Lemmas A.1 and A.2, we immediately obtain the following lemma.

Lemma A.3

Let $H_{j}\!\in \!\mathbb {C}$ and $H_{j}\ne \pm i$, $j=\overline{2,m}$. Then the functions $\Delta _{1}^{0}(\lambda )$ and $\Delta _{2}^{0}(\lambda )$ defined by (2.14) and (2.24), respectively, are sine-type functions of exponential type m.

Proof

Using Lemma A.1 and taking (3.7) into account, the functions $f_{1}(\lambda )$ and $f_{2}(\lambda )$ are sine-type functions of exponential type m. According to Lemma A.2 as well as (3.8), we arrive at the assertion. $\square$