Shrodinger算子在轴上的局部扰动

Dirac operator with linear potential and its perturbations

We prove that canonical Dirac expression with linear potential generates operators on axis and half axis, for which we can find the eigenvalues and eigenfunctions in explicit form. We construct the perturbations of these operators with in advance given spectra.

Counting function of magnetic eigenvalues for non-definite sign perturbations

We consider the perturbed operator $H(b,V) := H(b,0) + V$, where $H(b,0)$ is the $3$d Hamiltonian of Pauli with non-constant magnetic field, and $V$ is \textit{a non-definite sign electric potential} decaying exponentially with respect to the variable along the magnetic field. We prove that the only resonances of $H(b,V)$ near the low ground energy zero of $H(b,0)$ are its eigenvalues and are concentrated in the semi axis $(-\infty,0)$. Further, we establish new asymptotic expansions, upper and lower bounds on their number near zero.

周期Jacobi算子的反共振问题

Periodic Jacobi operator with finitely supported perturbations: the inverse resonance problem

我们考虑周期Jacobi算子$H$在${\Bbb Z}.$上的摄动。我们解决了逆共振问题：我们证明了从周期Jacobi算子$H$上的摄动到散射数据的映射：透射系数的逆和右半轴上的Jost函数，是一一对应的。我们考虑的散射数据的重建问题，从所有的本征值，共振和$R_-(λ)+1,$的零的集合，其中$R_-$是反射系数。