Semiclassical asymptotic spectrum of a Hartree-type operator near the upper boundary of spectral clusters

We consider the problem for eigenvalues of a perturbed two-dimensional oscillator in the case of a resonance frequency. The exciting potential is given by a Hartree-type integral operator with a smooth self-action potential. We find asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundary of spectral clusters, which form around energy levels of the nonperturbed operator. To calculate them, we use asymptotic formulas for quantum means.     

Lower bounds for the eigenvalues of the Spin~c Dirac-Witten operator

In this paper,we get optimal lower bounds for the eigenvalues of the Spin c Dirac-Witten operator.These estimates are given in terms of the mean curvature and different geometric invariants as the scalar curvature,the first eigenvalue of the perturbed Yamabe operator and the spinorial energy-momentum tensor.The limiting cases are also discussed.     

On the eigenvalues for a weighted p-Laplacian operator on metric graphs

We provide an explicit upper bound of the eigenvalues corresponding to a weighted p-Laplacian operator defined on a connected metric graph with finite total length.     

Subcritical perturbation of a locally periodic elliptic operator

We consider a singularly perturbed Dirichlet spectral problem for an elliptic operator of second order. The coefficients of the operator are assumed to be locally periodic and oscillating in the scale ? . We describe the leading terms of the asymptotics of the eigenvalues and the eigenfunctions to the problem, as the parameter ? tends to zero, under structural assumptions on the potential. More precisely, we assume that the local average of the potential has a unique global minimum point in the interior of the domain and its Hessian is non‐degenerate at this point. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotics of the eigenvalues and the formula for the trace of perturbations of the Laplace operator on the sphere \mathbb{S}^2

In this paper, we study the asymptotics of the eigenvalues of the Laplace operator perturbed by an arbitrary bounded operator on the sphere . For the first time, for the partial differential operator of second order, the leading term of the second correction of perturbation theory is obtained. A connection between the coefficient of the second term of the asymptotics of the eigenvalues and the formula for the traces of the operator under consideration is established.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 434–448Original Russian Text Copyright © 2005 by V. A. Sadovnichii, Z. Yu. Fazullin.This revised version was published online in April 2005 with a corrected issue number.     

Computation of eigenvalues of perturbed discrete semibounded operators

To compute the eigenvalues of a perturbed discrete semibounded operator, systems are obtained for the first time in which the number of equations is equal to the multiplicity of the corresponding eigenvalues of the unperturbed operator.     

On the spectrum of the Schrödinger operator perturbed by a rapidly oscillating potential

We study the spectrum of the one-dimensional Schrödinger operator perturbed by a rapidly oscillating potential. The oscillation period is a small parameter. We find explicitly the essential spectrum and study the existence of the discrete spectrum. Complete asymptotic expansions of the eigenvalues and corresponding eigenfunctions are constructed.     

The spectrum of the Schrödinger operator with a rapidly oscillating compactly supported potential

We study the effect of an eigenvalue appearing from the boundary of the essential spectrum of the Schrödinger operator perturbed by a rapidly oscillating compactly supported potential. We prove sufficient conditions for the existence and absence of such an eigenvalue and obtain the first few terms of its asymptotic expansion for the case where this eigenvalue exists.     

Weakly coupled bound states of Pauli operators

We consider the two-dimensional Pauli operator perturbed by a weakly coupled, attractive potential. We show that besides the eigenvalues arising from the Aharonov?CCasher zero modes there are two or one (depending on whether the flux of the magnetic field is integer or not) additional eigenvalues for arbitrarily small coupling and we calculate their asymptotics in the weak coupling limit.     

Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds

In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of the Dirichlet biharmonic operator on compact domains in a Euclidean space or a minimal submanifold of it and a unit sphere. We obtain universal bounds on the (k+1)th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains. The estimate for the (k+1)th eigenvalue of bounded domains in a Euclidean space improves an important inequality obtained recently by Cheng and Yang.     

On absence of embedded eigenvalues for schrÖdinger operators with perturbed periodic potentials

The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schrödinger operator with a periodic potential perturbed by a sufficiently fast decaying "impurity" potential. Results of this type have previously been known for the one-dimensional case only. Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. It is conjectured that all periodic potentials satisfy this condition. Separable periodic potentials satisfy it, and hence in dimensions two and three Schrödinger operator with a separable periodic potential perturbed by a sufficiently fast decaying "impurity" potential has no embedded eigenvalues     

Quasilevels of a two-particle Schrödinger operator with a perturbed periodic potential

We consider a two-dimensional periodic Schrödinger operator perturbed by the interaction potential of two one-dimensional particles. We prove that quasilevels (i.e., eigenvalues or resonances) of the given operator exist for a fixed quasimomentum and a small perturbation near the band boundaries of the corresponding periodic operator. We study the asymptotic behavior of the quasilevels as the coupling constant goes to zero. We obtain a simple condition for a quasilevel to be an eigenvalue.     

Decay law for a quasistationary state of the Schrödinger operator for a crystal film

For the Schrödinger operator in a cell corresponding to a crystal film pattern, eigenvalues may exist in the continuous spectrum and become resonances under perturbations. We prove that the corresponding decay law in a nonstationary approach is exponential for a nondegenerate (in some cases, degenerate) eigenvalue.     

Discrete Spectrum of the Schrodinger Operator Perturbed by a Narrowly Supported Potential

We study asymptotic properties of the discrete spectrum of the Schrodinger operator perturbed by a narrowly supported potential. The first terms of the asymptotic expansions in the small parameter equal to the width of the support of the potential are constructed for the eigenvalues and the corresponding eigenfunctions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 3, pp. 372–384, December, 2005.     

The levels of the two-particle Schrödinger operator corresponding to a crystal film

For a two-particle Schrödinger operator considered in a cell and having a potential periodic in four variables, we establish the existence of levels (i.e., eigenvalues or resonances) in the neighborhood of singular points of the unperturbed Green’s function and derive an asymptotic formula for these levels. We prove an existence and uniqueness theorem for the solution of the corresponding Lippmann-Schwinger equation.     

Perturbation of a periodic operator by a narrow potential

We consider perturbations of a second-order periodic operator on the line; the Schr?dinger operator with a periodic potential is a specific case of such an operator. The perturbation is realized by a potential depending on two small parameters, one of which describes the length of the potential support, and the inverse value of other corresponds to the value of the potential. We obtain sufficient conditions for the perturbing potential to have eigenvalues in the gaps of the continuous spectrum. We also construct their asymptotic expansions and present sufficient conditions for the eigenvalues of the perturbing potential to be absent.     

On Hill's operator with a matrix potential

In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the self‐adjoint operator generated by a system of Sturm–Liouville equations with summable coefficients and quasiperiodic boundary conditions. Then using these asymptotic formulas, we find conditions on the potential for which the number of gaps in the spectrum of the Hill's operator with matrix potential is finite. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类具抽象边界条件的迁移算子的谱分布

利用线性算子半群理论,研究了板几何中具抽象边界条件的各向异性、连续能量、非均匀介质的迁移方程．在假设边界算子日部分光滑和扰动算子K正则的条件下,采用豫解方法,得到了该迁移算子A的谱在区域Г中由至多可数个具有限代数重数的离散本征值组成等结果．     

Asymptotics of resonances for a Schrödinger operator with vector values

We show that the resonance counting function for a Schrödinger operator in dimension one has an asymptotic expansion and calculate an explicit expression for the leading term in some situations.     

Extrinsic estimates for eigenvalues of the Dirac operator

For a compact spin manifold M isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the square of the Dirac operator, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues. Since the unit sphere and the projective spaces admit the standard embedding into Euclidean spaces, we also obtain the corresponding results for their compact spin submanifolds.