On characteristic determinants of boundary value problems for Dirac type systems

A. Lunev ; M. Malamud

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 52, Tome 516 (2022), pp. 69-120

Résumé

The paper is concerned with the asymptotic behavior of the eigenvalues of the following $n \times n$ Dirac type equation $$ y' + Q(x) y = i \lambda B(x) y, y = \mathrm{col}\,(y_1, \ldots, y_n), x \in [0,\ell], $$ on a finite interval $[0,\ell]$ subject to general regular boundary conditions $C y(0) + D y(\ell) = 1$ with $C, D \in \mathbb{C}^{n \times n}$. Here $Q = (Q_{jk})_{j,k=1}^n$ is an integrable potential matrix and $ B = \mathrm{diag}\,(\beta_1, \ldots, \beta_n) = B^* \in L^1([0,\ell];\mathbb{R}^{n \times n}) $ is a diagonal matrix “weight”. If $n=2m$ and $ B(x) = \mathrm{diag}\,(-I_m, I_m) $ this equation is equivalent to $n\times n$ Dirac equation. Under the assumption $\mathrm{supp}\,(Q_{jk}) \subset \mathrm{supp}\,(\beta_k - \beta_j)$, we show that the deviation of the characteristic determinants $\Delta_Q(\cdot)$ and $\Delta_0(\cdot)$ of this boundary value problem (BVP) and the unperturbed BVP (with $Q \equiv 0$) is a Fourier transform of some integrable function, $$ \Delta_Q(\lambda) = \Delta_0(\lambda) + \int\limits_{\widetilde{b}_-}^{\widetilde{b}_+} g(u) e^{i \lambda u} du, g \in L^1[\widetilde{b}_-, \widetilde{b}_+]. $$ We apply this representation to study of zeros distribution of the characteristic determinant $\Delta_Q(\cdot)$ (eigenvalues of the above BPV) and show that $\Delta_Q(\cdot)$ is always an entire class $A$ function of exponential type, which is bounded on the real axis. We also find conditions guaranteeing that $\Delta_Q(\cdot)$ is a sine-type function and provide sharp asymptotic formula for its zeros. Finally, we show that if the entries of matrix $B(\cdot)$ can change sign within the segment $[0,\ell]$, then in general even in the case of regular boundary conditions eigenvalues split into two branches: the “good” branch lies in the horizontal strip and is close to the eigenvalues of the unperturbed BVP, while the “bad” branch has non-zero density and imaginary parts that tend to infinity. We illustrate this effect on a concrete $2 \times 2$ example.