Related papers: On Schr\"odinger operators with multisingular inve…
We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magnetic field which is described by the magnetic Schr\"odinger operator with a periodic potential plus a finitely supported perturbation. We describe all…
We present upper estimates for the number of negative eigenvalues of two-dimensional Schroedinger operators with potentials generated by Ahlfors regular measures of arbitrary dimension $\alpha\in (0, 2]$.The estimates are given in terms of…
In this work, we study inverse resonance problems for the Schr\"odinger operator on the real line with the potential supported in $[0,1]$. In general, all eigenvalues and resonances can not uniquely determine the potential. (i) It is shown…
We show that the absolute values of non-positive eigenvalues of Schr\"odinger operators with complex potentials can be bounded in terms of L_p-norms of the potential. This extends an inequality of Abramov, Aslanyan, and Davies to higher…
In this paper we obtain optimal multipolar Rellich inequality for biharmonic Schrodinger operator with positive multi-singular potentials. Moreover, we prove the attainability of the best constant and the criticality of the biharmonic…
New estimates for eigenvalues of non-self-adjoint multi-dimensional Schr\"{o}dinger operators are obtained in terms of $L_{p}$-norms of the potentials. The results extend and improve those obtained previously. In particular, diverse…
This paper deals with the approximation of a magnetic Schr\"odinger operator with a singular $\delta$-potential that is formally given by $(i \nabla + A)^2 + Q + \alpha \delta_\Sigma$ by Schr\"odinger operators with regular potentials in…
The paper presents estimates for the number of negative eigenvalues of a two-dimensional Schr\"odinger operator in terms of $L\log L$ type Orlicz norms of the potential and proves a conjecture by N.N. Khuri, A. Martin and T.T. Wu.
Schr\"odinger operators often display singularities at the origin, the Coulomb problem in atomic physics or the various matter coupling terms in the Friedmann-Robertson-Walker problem being prominent examples. For various applications it…
We study the distribution of the eigenvalues inside of the essential spectrum for discrete one-dimensional Schr\"odinger operators with potentials of Coulomb type decay.
We consider Schr\"odinger operators with a random potential which is the square of an alloy-type potential. We investigate their integrated density of states and prove Lifshits tails. Our interest in this type of models is triggered by an…
We consider the final state problem for the inhomogeneous nonlinear Schr\"odinger equation with a critical long-range nonlinearity. Given a prescribed asymptotic profile, which has a logarithmic phase correction compared with the free…
For $a \ge - {( \frac{{d}}{2}- 1)^2} $ and $2\sigma= {{d - 2}}-( {{{(d - 2)}^2} + 4a})^{1/2}$, let $$\begin{cases}\mathcal{H}_{a}= - \Delta + \frac{a} {{{{ | x |}^2}}},\\ \mathcal{\widetilde{H}}_{\sigma}= 2\big( { - \Delta + \frac{{{\sigma…
In this paper, we prove pure point spectrum for a large class of Schr\"odinger operators over circle maps with conditions on the rotation number going beyond the Diophantine. More specifically, we develop the scheme to obtain pure point…
We establish Anderson localization for 1-d discrete Schr\"odinger operators with positive weights. The distinctive feature of this work lies in the degeneracy of the weights, with both the potentials and weights assumed to be analytic and…
We consider the Schr\"odinger equation with a multipoint potential of the Bethe-Peierls-Thomas-Fermi type. We show that such a potential in dimension d=2 or d=3 is uniquely determined by its scattering amplitude at a fixed positive energy.…
We investigate existence and qualitative behaviour of solutions to nonlinear Schr\"odinger equations with critical exponent and singular electromagnetic potentials. We are concerned with magnetic vector potentials which are homogeneous of…
Starting from the semi-classical spectrum of Schr\"odinger operators $-h^2\Delta+V$ (on $\mathbb{R}^n$ or on a Riemannian manifold) it is possible to detect critical levels of the potential $V$. Via micro-local methods one can express…
We define the magnetic Schr\"odinger on an infinite graph by the data of a magnetic field, some weights on vertices and some weights on edges . We discuss essential self-adjointness of this operator for graphs of bounded degree. The main…
It is shown that the 1-dimensional lattice Schrodinger operator with potential given by a non-zero multiple of the Moebius function has positive Lyapounov exponent. In view of the classical theory of random Schrodinger operators, this may…