On the formula for characteristic determinants of boundary value problems for n × n Dirac type systems and its applications

Abstract

The paper is concerned with the spectral properties of the boundary value problems (BVP) associated with the following $n\times n$ Dirac type equation:$-i{y}^{\prime }-iQ\left(x\right)y=\lambda B\left(x\right)y,y=\mathrm{col}(y_1,\dots ,y_n),x\in [0,\ell ],$ on a finite interval $[0,\ell ]$ subject to the general two-point boundary conditions $Cy\left(0\right)+Dy\left(\ell \right)=0$ with $C,D\in C^{n\times n}$. Here $Q={\left(Q_{jk}\right)}_{j,k=1}^n$ is an integrable potential matrix and $B=\mathrm{diag}({\beta }_1,\dots ,{\beta }_n)=B^{⁎}$ is a diagonal integrable matrix “weight”. If $n=2m$ and $B(\cdot )=\mathrm{diag}(-I_m,I_m)$, this equation turns into $n\times n$ Dirac equation.

First, assuming that $\mathrm{supp}\left(Q_{jk}\right)\subset \mathrm{supp}({\beta }_k-{\beta }_j)$, we show that the deviation ${\Phi }_Q(\cdot ,\lambda )-{\Phi }_0(\cdot ,\lambda )$ of the fundamental matrix solutions to the above perturbed and unperturbed ($Q=0$) equation is represented as a Fourier transform of a certain matrix kernel $K_Q(\cdot ,\cdot )$ from a special Banach space.

This result is used to prove our main result, the following formula for the deviation of the characteristic determinants ${\Delta }_Q(\cdot )$ and ${\Delta }_0(\cdot )$ of two (perturbed and unperturbed) BVPs as a Fourier transform,${\Delta }_Q\left(\lambda \right)={\Delta }_0\left(\lambda \right)+\underset{b_{-}}{\overset{b_{+}}{\int }}g\left(u\right)e^{i\lambda u}du,g\in L^1[b_{-},b_{+}],$ where $b_{\pm }$ are explicitly expressed via entries of the matrix function $B(\cdot )$.

In turn, assuming that each function ${\beta }_k(\cdot )$ is of fixed sign (so-called “fixed sign” condition), this representation yields asymptotic behavior of the spectrum in the case of regular boundary conditions. Namely, we show that ${\lambda }_m={\lambda }_m^0+o\left(1\right)$ as $m\to \infty$, where ${\left\{{\lambda }_m\right\}}_{m\in Z}$ and ${\left\{{\lambda }_m^0\right\}}_{m\in Z}$ are sequences of eigenvalues of perturbed and unperturbed BVP, respectively. It is also shown that for $Q\in L^p$, $p\in (1,2]$, the following estimate holds under additional condition on $B(\cdot )$:$\sum _{m\in Z}|{\lambda }_m-{\lambda }_m^0{|}^{p^{\prime }}+\sum _{m\in Z}{(1+|m\left|\right)}^{p-2}|{\lambda }_m-{\lambda }_m^0{|}^p<\infty ,p^{\prime }:=p/(p-1).$

In the case of Dirac operator, we show that the sequence of the eigenvalues splits into the union of n branches asymptotically close to arithmetic progressions ${\{2\pi k-i\mathrm{Log}(-{\mu }_s\left)\right\}}_{k\in Z}$, $s\in \{1,\dots ,n\}$, where ${\mu }_1,\dots ,{\mu }_n$ are the eigenvalues of a certain $n\times n$ matrix explicitly constructed via the matrices C and D.