相关论文: Gap of the First Two Eigenvalues of the Schr\"odin…
We prove Strichartz estimates for the absolutely continuous evolution of a Schr\"odinger operator $H = (i\nabla + A)^2 + V$ in $\R^n$, $n > 2$. Both the magnetic and electric potentials are time-independent and satisfy pointwise polynomial…
We prove quantitative bounds on the eigenvalues of non-selfadjoint unbounded operators obtained from selfadjoint operators by a perturbation that is relatively-Schatten. These bounds are applied to obtain new results on the distribution of…
Dealing with one-dimensional diffusion operators, we obtain upper and lower variational formulae on the eigenvalues given by the max-min principle, generalizing the celebrated result of Chen and Wang on the spectral gap. Our inequalities…
It is known that the spectrum of Schr\"odinger operators with sparse potentials consists of singular continuous spectrum. We give a sufficient condition so that the edge of the singular continuous spectrum is not an eigenvalue and construct…
We consider tunneling between symmetric wells for a 2-D semi-classical Schr\"odinger operator for energies close to the quadratic minimum of the potential V in two cases: (1) excitations of the lowest frequency in the harmonic oscillator…
We present, to the best of our knowledge, the first numerical algorithm for explicit, computable two-sided eigenvalue bounds for Schr\"odinger operators H = -Delta + V on R^N, N = 2,3, in the presence of both an unbounded potential and an…
In this paper we study a connection between finite-gap on one energy level two-dimensional Schrodinger operators and two-dimensional discrete operators. We find spectral data for a new class of two-dimensional integrable discrete operators.…
The motivation of this paper is to study a second order elliptic operator which appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant $r$-mean curvature. We prove a generalized Bochner-type…
The phenomenon "hypo-coercivity," i.e., the increased rate of contraction for a semi-group upon adding a large skew-adjoint part to the generator, is considered for 1D semigroups generated by the Schr\"odinger operators $-\partial^2_x + x^2…
We consider a quantum particle moving in the one dimensional lattice Z and interacting with a indefinite sign external field v. We prove that the associated discrete Schroedinger operator H can have one or two eigenvalues, situated as below…
For bounded linear operators $A,B$ on a Hilbert space $\mathcal{H}$ we show the validity of the estimate $$ \sum_{\lambda \in \sigma_d (B)} \dist(\lambda, \overline{\num}(A))^p \leq \| B-A \|_{\mathcal{S}_p}^p$$ and apply it to recover and…
We consider the Schrodinger operator a given domain. Our goal is to study some optimization problems where an optimal (non-negative) potential V has to be determined in some suitable admissible classes and for some suitable optimization…
Simon's results on the negative spectrum of recurrent Schr\"{o}dinger operators ($d=1,2$) are extended to a wider class of potentials and to non-local operators. An example of $L^1-$potental is constructed for which the essential spectrum…
Let L be a Schroedinger operator with potential W in L^{(n+1)/2}. We prove that there is no embedded eigenvalue. The main tool is an Lp Carleman type estimate, which builds on delicate dispersive estimates established in a previous paper.…
We prove the uniform lower bound for the difference $\lambda_2 - \lambda_1$ between first two eigenvalues of the fractional Schr\"odinger operator, which is related to the Feynman-Kac semigroup of the symmetric $\alpha$-stable process…
We review some results and proofs on eigenvalue bounds for random Schr\"odinger operators with complex-valued potentials. We also include new Schatten norm estimates for the resolvent and use them to obtain bounds for sums of eigenvalues.
We study the distribution of eigenvalues of the Schr\"odinger operator with a complex valued potential $V$. We prove that if $|V|$ decays faster than the Coulomb potential, then all eigenvalues are in a disc of a finite radius.
In this paper we obtain asymptotic formulas of arbitrary order for the Bloch eigenvalue and the Bloch function of the periodic Schrodinger operator of arbitrary dimension, when corresponding quasimomentum lies near a diffraction hyperplane.…
We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator $- 4 d^2/ds^2 + \kappa^2 (s)$ with potential given by the curvature of a closed curve.
We study location of eigenvalues of one-dimensional discrete Schr\"odinger operators with complex $\ell^{p}$-potentials for $1\leq p\leq \infty$. In the case of $\ell^{1}$-potentials, the derived bound is shown to be optimal. For $p>1$, two…