Isospectral Dirac operators

TL;DR

This paper characterizes self-adjoint Dirac operators on the interval [0, π] that share identical spectra, providing insights into their spectral properties and classifications.

Contribution

It offers a description of all self-adjoint regular Dirac operators on [0, π] with the same spectrum, advancing spectral theory understanding.

Findings

Characterization of isospectral Dirac operators

Conditions for operators to share spectra

Spectral classification results

Abstract

We give the description of self-adjoint regular Dirac operators, on $[0, \pi]$, with the same spectra.

Equations

$$\ell y\equiv \left\{B\frac{d}{dx}+\Omega (x)\right\}y$$

$$y_1(0)\cos \alpha +y_2(0)\sin \alpha$$

$$y_1(\pi )$$

$$B=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),\Omega (x)=\left(\begin{array}{cc}p(x)& q(x)\\ q(x)& -p(x)\end{array}\right).$$

$$D=$$

$$\displaystyle\phantom{\bigg{\{}}y_{1}(0)\cos\alpha+y_{2}(0)\sin\alpha=0,\quad y_{1}(\pi)=0\bigg{\}}$$

$${\lambda }_n(\Omega ,\alpha )=n-\frac{\alpha }{\pi }+r_n,r_n=o(1),n\to \pm \infty .$$

$$\sum _{n=-\infty }^{\infty }{r}_n^2<\infty .$$

$$\ell \phi =\lambda \phi ,\phi (0,\lambda )=\left(\begin{array}{c}\sin \alpha \\ -\cos \alpha \end{array}\right).$$

$$a_n=\Vert {\phi }_n{\Vert }^2={\int }_0^{\pi }|{\phi }_n(x,\Omega ){|}^{2}dx,n\in \mathbb{Z}.$$

$$h_n(x,\Omega )=h_n(x)=\frac{{\phi }_n(x,\Omega )}{\sqrt{a_n(\Omega ,\alpha )}}.$$

$$a_n(\Omega )=\pi +c_n,\sum _{n=-\infty }^{\infty }{c}_n^2<\infty .$$

$$\frac{\alpha }{\pi }=\underset{n\to \infty }{\lim }(n-{\lambda }_n(\Omega ,\alpha ))=\underset{n\to \infty }{\lim }(n-{\lambda }_n(\tilde{\Omega },\tilde{\alpha }))=\frac{\tilde{\alpha }}{\pi }.$$

$$(\Omega ,\alpha )\in L_{\mathbb{R}}^2[0,\pi ]\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right]⟷\{{\lambda }_n(\Omega ,\alpha ),a_n(\Omega ,\alpha );n\in \mathbb{Z}\}$$

$$M^2(\Omega )=\{\tilde{\Omega }\in L_{\mathbb{R}}^2[0,\pi ]:{\lambda }_n(\tilde{\Omega },\tilde{\alpha })={\lambda }_n(\Omega ,\alpha ),n\in \mathbb{Z}\}.$$

$$\tilde{\Omega }\in M^2(\Omega )\leftrightarrow \{a_n(\tilde{\Omega }),n\in \mathbb{Z}\}$$

$$e^{t_n}=1+d_n,\sum _{n=-\infty }^{\infty }{d}_n^2<\infty .$$

$$\tilde{\Omega }\in M^2(\Omega )\leftrightarrow \{t_n(\tilde{\Omega }),n\in \mathbb{Z}\}\in l^2$$

$$\Omega (x,t)=\Omega (x)+\frac{e^t-1}{{\theta }_m(x,t,\Omega )}\{B{h}_m(x,\Omega )h_m^{\ast }(x,\Omega )-h_m(x,\Omega )h_m^{\ast }(x,\Omega )B\},$$

$$h_n(x,\Omega (\cdot ,t))=\{\begin{array}{ll}\frac{e^{-t/2}}{{\theta }_m(x,t,\Omega )}{h}_m(x,\Omega ),& \text{mbox}ifn=m,\\ h_n(x,\Omega )-\frac{(e^t-1){\int }_0^x{h}_m^{\ast }(s,\Omega )h_n(s,\Omega )ds}{{\theta }_m(x,t,\Omega )}{h}_m(x,\Omega ),& \text{mbox}ifn\ne m.\end{array}$$

$$\tilde{\Omega }(x)={\Omega }_{m,t}(x)=\frac{\pi (e^t-1)}{\pi +(e^t-1)x}\left(\begin{array}{cc}-\sin 2mx& \cos 2mx\\ \cos 2mx& \sin 2mx\end{array}\right),$$

$$T_{-1}$$

$$T_0$$

$$T_1$$

$$T_2$$

$$\text{mathmakebox}[\text{widthof}=][c]⋮$$

$$T_{2n}$$

$$T_{2n+1}$$

$$\text{mathmakebox}[\text{widthof}=][c]⋮$$

$$\Omega (x,T_m)=\Omega (x,T_{m-1})+△\Omega (x,T_m),m=0,1,2,\dots ,$$

$$△\Omega (x,T_m)=\frac{e^{t_{\tilde{m}}}-1}{{\theta }_m(x,t_{\tilde{m}},\Omega (\cdot ,T_{m-1}))}[B{h}_{\tilde{m}}(x,\Omega (\cdot ,T_{m-1}))h_{\tilde{m}}^{\ast }(\cdot )-h_{\tilde{m}}(\cdot )h_{\tilde{m}}^{\ast }(\cdot )B],$$

$$\Omega (x,T)\equiv \Omega (x)+\sum _{m=0}^{\infty }△\Omega (x,T_m)\in M^2(\Omega ).$$

$$N(T_0)$$

$$N(T_1)$$

$$N(T_2)$$

$$\text{mathmakebox}[\text{widthof}=][c]⋮$$

$$N(T_{2n})$$

Full text

Isospectral Dirac operators

Yuri Ashrafyan

Yerevan State University, Alex Manoogian 1, Yerevan, 0025, Armenia

Tigran Harutyunyan111Corresponding author. Email: hartigr@yahoo.co.uk

Yerevan State University, Alex Manoogian 1, Yerevan, 0025, Armenia

Received 30 December 2015, appeared 23 January 2017

Communicated by Miklós Horváth

Abstract. We give the description of self-adjoint regular Dirac operators, on $[0,\pi]$, with the same spectra.

Keywords: inverse spectral theory, Dirac operator, isospectral operators.

2010 Mathematics Subject Classification: 34A55, 34B30, 47E05.

1 Introduction and statement of result

Let $p$ and $q$ are real-valued, summable on $[0,\pi]$ functions, i.e. $p,q\in L^{1}_{\mathbb{R}}[0,\pi]$. By $L(p,q,\alpha)=L(\Omega,\alpha)$ we denote the boundary-value problem for canonical Dirac system (see [6, 7, 10, 14, 15]):

[TABLE]

where

[TABLE]

By the same $L(p,q,\alpha)$ we also denote a self-adjoint operator generated by differential expression $\ell$ in Hilbert space of two component vector-function $L^{2}([0,\pi];{\mathbb{C}}^{2})$ on the domain

[TABLE]

where $AC[0,\pi]$ is the set of absolutely continuous functions on $[0,\pi]$ (see, e.g. [14, 17]). It is well known (see [2, 6, 10]) that under these conditions the spectra of the operator $L(p,q,\alpha)$ is purely discrete and consists of simple, real eigenvalues, which we denote by $\lambda_{n}=\lambda_{n}(p,q,\alpha)=\lambda_{n}(\Omega,\alpha)$, $n\in\mathbb{Z}$, to emphasize the dependence of $\lambda_{n}$ on quantities $p,q$ and $\alpha$. It is also well known (see, e.g. [2, 6, 10]) that the eigenvalues form a sequence, unbounded below as well as above. So we will enumerate it as $\lambda_{k}<\lambda_{k+1},k\in\mathbb{Z}$, $\lambda_{k}>0$, when $k>0$ and $\lambda_{k}<0$, when $k<0$, and the nearest to zero eigenvalue we will denote by $\lambda_{0}$. If there are two nearest to zero eigenvalue, then by $\lambda_{0}$ we will denote the negative one. With this enumeration it is proved (see [2, 6, 10]), that the eigenvalues have the asymptotics:

[TABLE]

In what follows, writing $\Omega\in A$ will mean $p,q\in A$. If $\Omega\in L^{2}_{\mathbb{R}}[0,\pi]$, then we know, (see, e.g. [10]), that instead of $r_{n}=o(1)$ we have:

[TABLE]

Let $\varphi(x,\lambda)=\varphi(x,\lambda,\alpha,\Omega)$ be the solution of the Cauchy problem

[TABLE]

Since the differential expression $\ell$ self-adjoint, then the components $\varphi_{1}(x,\lambda)$ and $\varphi_{2}(x,\lambda)$ of the vector-function $\varphi(x,\lambda)$ we can choose real-valued for real $\lambda$. By $a_{n}=a_{n}(\Omega,\alpha)$ we denote the squares of the $L^{2}$-norm of the eigenfunctions $\varphi_{n}(x,\Omega)=\varphi(x,\lambda_{n}(\Omega,\alpha),\alpha,\Omega)$:

[TABLE]

The numbers $a_{n}$ are called norming constants. And by $h_{n}(x,\Omega)$ we will denote normalized eigenfunctions (i.e. $\|h_{n}(x)\|=1$):

[TABLE]

It is known (see [6, 10]) that in the case of $\Omega\in L^{2}_{\mathbb{R}}[0,\pi]$ the norming constants have an asymptotic form:

[TABLE]

Definition 1.1**.**

Two Dirac operators $L(\Omega,\alpha)$ and $L(\tilde{\Omega},\tilde{\alpha})$ are said to be isospectral, if $\lambda_{n}(\Omega,\alpha)=\lambda_{n}(\tilde{\Omega},\tilde{\alpha})$, for every $n\in\mathbb{Z}$.

Lemma 1.2**.**

Let $\Omega,\tilde{\Omega}\in L^{1}_{\mathbb{R}}[0,\pi]$ and the operators $L(\Omega,\alpha)$ and $L(\tilde{\Omega},\tilde{\alpha})$ are isospectral. Then $\tilde{\alpha}=\alpha$.

Proof.

The proof follows from the asymptotics (1.4):

[TABLE]

∎

So, instead of isospectral operators $L(\Omega,\alpha)$ and $L(\tilde{\Omega},\tilde{\alpha})$, we can talk about “isospectral potentials” $\Omega$ and $\tilde{\Omega}$.

Theorem 1.3** (Uniqueness theorem).**

The map

[TABLE]

is one-to-one.

Remark 1.4**.**

It is natural to call this a Marchenko theorem, since it is an analogue of the famous theorem of V. A. Marchenko [16], in the case for Sturm–Liouville problem. The proof of this theorem for the case $p,q\in AC[0,\pi]$ there is in the paper [19]. The detailed proof for the case $p,q\in L^{2}_{\mathbb{R}}[0,\pi]$ there is in [8] (see also [5, 6, 7, 9, 11, 20]).

Let us fix some $\Omega\in L^{2}_{\mathbb{R}}[0,\pi]$ and consider the set of all canonical potentials $\tilde{\Omega}=\begin{pmatrix}\tilde{p}&\tilde{q}\\ \tilde{q}&-\tilde{p}\\ \end{pmatrix}$, with the same spectra as $\Omega$:

[TABLE]

Our main goal is to give the description of the set $M^{2}(\Omega)$ as explicit as it possible.

From the uniqueness theorem the next corollary easily follows.

Corollary 1.5**.**

The map

[TABLE]

is one-to-one.

Since $\tilde{\Omega}\in M^{2}(\Omega)$, then $a_{n}(\tilde{\Omega})$ have similar to (1.8) asymptotics. Since $a_{n}(\Omega)$ and $a_{n}(\tilde{\Omega})$ are positive numbers, there exist real numbers $t_{n}=t_{n}(\tilde{\Omega})$, such that $\frac{a_{n}(\Omega)}{a_{n}(\tilde{\Omega})}=e^{t_{n}}$. From the latter equality and from (1.8) follows that

[TABLE]

It is easy to see, that the sequence $\{t_{n};\,n\in\mathbb{Z}\}$ is also from $l^{2}$, i.e. $\sum_{n=-\infty}^{\infty}t_{n}^{2}<\infty$. Since all $a_{n}(\Omega)$ are fixed, then from the corollary 1.5 and the equality $a_{n}(\tilde{\Omega})=a_{n}(\Omega)e^{-t_{n}}$ we will get the following corollary.

Corollary 1.6**.**

The map

[TABLE]

is one-to-one.

Thus, each isospectral potential is uniquely determined by a sequence $\{t_{n};\,n\in\mathbb{Z}\}$. Note, that the problem of description of isospectral Sturm–Liouville operators was solved in [4, 12, 13, 18].

For Dirac operators the description of $M^{2}(\Omega)$ is given in [9]. This description has a “recurrent” form, i.e. at the first in [9] is given the description of a family of isospectral potentials $\Omega(x,t),\ t\in\mathbb{R}$, for which only one norming constant $a_{m}(\Omega(\cdot,t))$ different from $a_{m}(\Omega)$ (namely, $a_{m}(\Omega(\cdot,t))=a_{m}(\Omega)e^{-t}$), while the others are equal, i.e. $a_{m}(\Omega(\cdot,t))=a_{m}(\Omega)$, when $n\neq m$.

Theorem 1.7** ([9]).**

Let $t\in\mathbb{R}$, \alpha\in\big{(}-\frac{\pi}{2},\frac{\pi}{2}\big{]} and

[TABLE]

where $\theta_{n}(x,t,\Omega)=1+(e^{t}-1)\int_{0}^{x}|h_{n}(s,\Omega)|^{2}ds$, and $*$ is a sign of transponation, e.g. $h_{m}^{*}=\begin{pmatrix}h_{m_{1}}\\ h_{m_{2}}\\ \end{pmatrix}^{*}=(h_{m_{1}},h_{m_{2}})$. Then, for arbitrary $t\in\mathbb{R}$, $\lambda_{n}(\Omega,t)=\lambda_{n}(\Omega)$ for all $n\in\mathbb{Z}$, $a_{n}(\Omega,t)=a_{n}(\Omega)$ for all $n\in\mathbb{Z}\backslash\{m\}$ and $a_{m}(\Omega,t)=a_{m}(\Omega)e^{-t}$. The normalized eigenfunctions of the problem $L(\Omega(\cdot,t),\alpha)$ are given by the formulae:

[TABLE]

Theorem 1.7 shows that it is possible to change exactly one norming constant, keeping the others. As examples of isospectral potentials $\Omega$ and $\tilde{\Omega}$ we can present $\Omega(x)\equiv 0=\begin{pmatrix}0&0\\ 0&0\\ \end{pmatrix}$ and

[TABLE]

where $t\in\mathbb{R}$ is an arbitrary real number and $m\in\mathbb{Z}$ is an arbitrary integer.

Changing successively each $a_{m}(\Omega)$ by $a_{m}(\Omega)e^{-t_{m}}$, we can obtain any isospectral potential, corresponding to the sequence $\{t_{m};m\in\mathbb{Z}\}\in l^{2}$. It follows from the uniqueness Theorem 1.3 that the sequence, in which we change the norming constants, is not important.

In [9] were used the following designations:

[TABLE]

Let $\Omega(x,T_{-1})\equiv\Omega(x)$ and

[TABLE]

where

[TABLE]

where $\tilde{m}=\frac{m+1}{2}$, if $m$ is odd and $\tilde{m}=-\frac{m}{2}$, if $m$ is even. The arguments in others $h_{\tilde{m}}(\cdot)$ and $h_{\tilde{m}}^{*}(\cdot)$ are the same as in the first. And after that in [9] the following theorem was proved.

Theorem 1.8** ([9]).**

Let $T=\{t_{n},n\in\mathbb{Z}\}\in l^{2}$ and $\Omega\in L^{2}_{\mathbb{R}}[0,\pi]$. Then

[TABLE]

We see, that each potential matrix $\bigtriangleup\Omega(x,T_{m})$ defined by normalized eigenfunctions $h_{\tilde{m}}(x,\Omega(x,T_{m-1}))$ of the previous operator $L(\Omega(\cdot,T_{m-1}),\alpha)$. This approach we call “recurrent” description.

In this paper, we want to give a description of the set $M^{2}(\Omega)$ only in terms of eigenfunctions $h_{n}(x,\Omega)$ of the initial operator $L(\Omega,\alpha)$ and sequence $T\in l^{2}$. With this aim, let us denote by $N(T_{m})$ the set of the positions of the numbers in $T_{m}$, which are not necessary zero, i.e.

[TABLE]

in particular $N(T)\equiv\mathbb{Z}$. By $S(x,T_{m})$ we denote the $(m+1)\times(m+1)$ square matrix

[TABLE]

where $\delta_{ij}$ is a Kronecker symbol. By $S_{p}^{(k)}(x,T_{m})$ we denote a matrix which is obtained from the matrix $S(x,T_{m})$ by replacing the $k$th column of $S(x,T_{m})$ by $H_{p}(x,T_{m})\!=\!\{-(e^{t_{k}}-1)h_{k_{p}}(x)\}_{k\in N(T_{m})}$ column, $p=1,2$, Now we can formulate our result as follows.

Theorem 1.9**.**

Let $T=\{t_{k}\}_{k\in\mathbb{Z}}\in l^{2}$ and $\Omega\in L^{2}_{\mathbb{R}}[0,\pi]$. Then the isospectral potential from $M^{2}(\Omega)$, corresponding to $T$, is given by the formula

[TABLE]

where

[TABLE]

and $\det S(x,T)=\lim_{m\rightarrow\infty}\det S(x,T_{m})$ (the same for $\det S_{p}^{k}(x,T),\ p=1,2$).

In addition, for $p(x,T)$ and $q(x,T)$ we get explicit representations:

[TABLE]

2 Proof of Theorem 1.9

The spectral function of an operator $L(\Omega,\alpha)$ defined as

[TABLE]

i.e. $\rho(\lambda)$ is left-continuous, step function with jumps in points $\lambda=\lambda_{n}$ equals $\frac{1}{a_{n}}$ and $\rho(0)=0$.

Let $\Omega,\tilde{\Omega}\in L^{2}_{\mathbb{R}}[0,\pi]$ and they are isospectral. It is known (see [2, 3, 7, 14]), that there exists a function $G(x,y)$ such that:

[TABLE]

It is also known (see, e.g. [2, 7, 14]), that the function $G(x,y)$ satisfies to the Gelfand–Levitan integral equation:

[TABLE]

where

[TABLE]

If the potential $\tilde{\Omega}$ from $M^{2}(\Omega)$ is such that only finite norming constants of the operator $L(\tilde{\Omega},\alpha)$ are different from the norming constants of the operator $L(\Omega,\alpha)$, i.e. $a_{n}(\tilde{\Omega})=a_{n}(\Omega)e^{-t_{n}},\ n\in N(T_{m})$ and the others are equal, then it means, that

[TABLE]

where $\delta$ is Dirac $\delta$-function. In this case the kernel $F(x,y)$ can be written in a form of a finite sum (using notation (1.7)):

[TABLE]

and consequently, the integral equation (2.2) becomes to an integral equation with degenerated kernel, i.e. it becomes to a system of linear equations and we will look for the solution in the following form:

[TABLE]

where $g_{k}(x)=\begin{pmatrix}g_{k_{1}}(x)\\ g_{k_{2}}(x)\\ \end{pmatrix}$ is an unknown vector-function. Putting the expressions (2.5) and (2.6) into the integral equation (2.2) we will obtain a system of algebraic equations for determining the functions $g_{k}(x)$:

[TABLE]

where

[TABLE]

It would be better if we consider the equations (2.7) for the vectors $g_{k}=\begin{pmatrix}g_{k_{1}}\\ g_{k_{2}}\\ \end{pmatrix}$ by coordinates $g_{k_{1}}$ and $g_{k_{2}}$ to be a system of scalar linear equations:

[TABLE]

The systems (2.8) might be written in matrix form

[TABLE]

where the column vectors $g_{p}(x,T_{m})=\{g_{k_{p}}(x,T_{m})\}_{k\in N(T_{m})},\ p=1,2$, and the solution can be found in the form (Cramer’s rule):

[TABLE]

Thus we have obtained for $g_{k}(x)$ the following representation:

[TABLE]

and then by putting (2.10) into (2.6) we find the $G(x,y,T_{m})$ function. If the potential $\Omega$ is from $L^{1}_{\mathbb{R}}$, then such is also the kernel $G(x,x,T_{m})$ (see [9]), and the relation between them gives as follows:

[TABLE]

On the other hand we have

[TABLE]

So, using the Theorem 1.8 and the equality (2.12) we can pass to the limit in (2.11), when $m\rightarrow\infty$:

[TABLE]

The potentials $\Omega(x,T)$ in (1.10) and (2.13) have the same spectral data $\{\lambda_{n}(T),a_{n}(T)\}_{n\in\mathbb{Z}}$, and therefore they are the same and $\Omega(\cdot,T)$ defined by (2.13) is also from $M^{2}(\Omega)$.

Using (2.6) and (2.10) we calculate the expression $G(x,x,T_{m})B-BG(x,x,T_{m})$ and pass to the limit, obtaining for the $p(x,T)$ and $q(x,T)$ the representations:

[TABLE]

Theorem 1.9 is proved.

For example, when we change just one norming constant (e.g. for $T_{0}$) we get two independent linear equations:

[TABLE]

For the solutions we get:

[TABLE]

and for the potentials $p(x,T_{0})$ and $q(x,T_{0})$:

[TABLE]

Acknowledgements

This work was supported by the RA MES State Committee of Science, in the frames of the research project No.15T-1A392.