Related papers: Compactness in the d-bar Neumann problem, magnetic…
We study first the Hamiltonian operator H corresponding to the Fock-Weyl extension of the Dirac equation to gravitation. When searching for stationary solutions to this equation, in a static metric, we show that just one invariant Hermitian…
We consider a self-adjoint two-dimensional Schr\"odinger operator $H_{\alpha\mu}$, which corresponds to the formal differential expression \[ -\Delta - \alpha\mu, \] where $\mu$ is a finite compactly supported positive Radon measure on…
In this paper, we investigate the behavior of the eigenvalues of a magnetic Aharonov-Bohm operator with half-integer circulation and Dirichlet boundary conditions in a bounded planar domain. We establish a sharp relation between the rate of…
We study the time-dependent Aharonov-Bohm effect on the noncommutative space. Because there is no net Aharonov-Bohm phase shift in the time-dependent case on the commutative space, therefore, a tiny deviation from zero indicates new…
It is common that a Sobolev space defined on $\mathbb{R}^m$ has a non-compact embedding into an $L^p$-space, but it has subspaces for which this embedding becomes compact. There are three well known cases of such subspaces, the Rellich…
We consider the Schr\"odinger operator $H$ with a periodic potential $p$ plus a compactly supported potential $q$ on the half-line. We prove the following results: 1) a forbidden domain for the resonances is specified, 2) asymptotics of the…
In a seminal work, S.A.R. Horsley and collaborators [S.A.R. Horsley {\em et al.}, Nature Photon. {\bf 9}, 436 (2015)] have shown that, in the framework of non-Hermitian extensions of the Schr\"odinger and Helmholtz equations, a localized…
We prove compactness of a restricted set of real-valued, compactly supported potentials $V$ for which the corresponding Schr\"odinger operators $H_V$ have the same resonances, including multiplicities. More specifically, let $B_R(0)$ be the…
In this work the bound state and scattering problems for a spin-1/2 particle undergone to an Aharonov-Bohm potential in a conical space in the nonrelativistic limit are considered. The presence of a \delta-function singularity, which comes…
We study sufficient conditions for the absence of positive eigenvalues of magnetic Schr\"odinger operators in $\mathbb{R}^d,\, d\geq 2$. In our main result we prove the absence of eigenvalues above certain threshold energy which depends…
Nonlinear Schrodinger Equations (NLS) of the Hartree type occur in the modeling of quantum semiconductor devices. Their "semiclassical" limit of vanishing (scaled) Planck constant is both a mathematical challenge and practically relevant…
For a bounded weak Lipschitz domain we show the so called `Maxwell compactness property', that is, the space of square integrable vector fields having square integrable weak rotation and divergence and satisfying mixed tangential and normal…
We will discuss the asymptotic behaviour of the eigenvalues of Schr\"{o}dinger operator with a matrix potential defined by Neumann boundary condition in $L_2^m(F)$, where $F$ is $d$-dimensional rectangle and the potential is a $m \times m$…
Compactons are studied in the framework of the Korteweg-de Vries (KdV) equation with the sublinear nonlinearity. Compactons represent localized bell-shaped waves of either polarity which propagate to the same direction as waves of the…
We discuss the Hamiltonian for a nonrelativistic electron with spin in the presence of an abelian magnetic monopole and note that it is not self-adjoint in the lowest two angular momentum modes. We then use von Neumann's theory of…
We consider the Schr\"odinger operator with constant magnetic field defined on the half-plane with a Dirichlet boundary condition, $H_0$, and a decaying electric perturbation $V$. We analyze the spectral density near the Landau levels,…
On a bounded smooth pseudoconvex domain in $\mathbb{C}^n$ with $n >2$, inspired by the compactness condition introduced by Yue Zhang, we present the new sufficient condition for the exact regularity of the $\overline\partial$-Neumann…
We search for anomalous normal and superconductive behavior in the two-band Hubbard model with one narrow band. We analyze the influence of electron-polaron effect and Altshuler-Aronov effect on effective mass enhancement and scattering…
Some frustrated magnets and superconducting arrays possess unusual symmetries that cause the free energy or other physics of a $D$-dimensional quantum or classical problem to be that of a different problem in a reduced dimension $d<D$.…
We study composition operators on the Schwartz space of rapidly decreasing functions. We prove that such a composition operator is never a compact operator and we obtain necessary or sufficient conditions for the range of the composition…