Speaker
Description
In this talk it will be discussed spectral properties of boundary value problems for the following first order system of ordinary differential equations
\begin{equation} %\label{eq:Ly.abstract}
L y = -i B(x)^{-1} \bigl(y' + Q(x) y\bigr) = \lambda y , \quad
B(x) = B(x)^, \quad y= {\rm co}l(y_1, \ldots, y_n), \quad x \in [0,\ell],
\end{equation*}
on a finite interval $[0,\ell]$. Here $Q \in L^1([0,\ell]; \bC^{n \times n})$ is a potential matrix and $B \in L^{\infty}([0,\ell]; \bR^{n \times n})$ is an invertible self-adjoint diagonal ``weight'' matrix. If $n=2m$ and $B(x) = {\rm diag}(-I_m, I_m)$ this equation is equivalent to the classical Dirac equation of order $n$. We will discuss the spectral properties of boundary value problems associated with the above equation subject to general boundary conditions $U(y)=Cy(0)+Dy(\ell) = 0$ satisfying the maximality condition ${\rm rank}(C \ D) = n$.
One of our main results is the formula for the deviation of the characteristic determinants $\Delta(\lambda) - \Delta_0(\lambda)$ of the perturbed and unperturbed (with $Q=0$) boundary value problems subject to the same boundary conditions $U(y) = 0$. Namely it is shown that it is a Fourier transform of a certain integrable function explicitly expressed via kernels of the transformation operators.
In turn, this result is applied to establish asymptotic behavior of eigenvalues as well as sharp evaluation of the reminder.
Joint work with Anton Lunyov.
