A trace formula and high-energy spectral asymptotics for the perturbed Landau Hamiltonian

A two-dimensional Schrödinger operator with a constant magnetic field perturbed by a smooth compactly supported potential is considered. The spectrum of this operator consists of eigenvalues which accumulate to the Landau levels. We call the set of eigenvalues near the nth Landau level an nth eigenvalue cluster, and study the distribution of eigenvalues in the nth cluster as n→∞. A complete asymptotic expansion for the eigenvalue moments in the nth cluster is obtained and some coefficients of this expansion are computed. A trace formula involving the eigenvalue moments is obtained.     

Manifolds with small Dirac eigenvalues are nilmanifolds

Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and bounded diameter, and almost non-negative scalar curvature. Let r = 1 if n = 2,3 and r = 2^[n/2]-1 + 1 if n ≥ 4. We show that if the square of the Dirac operator on such a manifold has r small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with almost non-negative scalar curvature has one small Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on M with a parallel spinor. In dimension 4 this implies that M is either a torus or a K3-surface.      

The Second Dirac Eigenvalue of a Nearly Parallel G2-Manifold

We investigate the second Dirac eigenvalue on Riemannian manifolds admitting a Killing spinor. In small dimensions the whole Dirac spectrum depends on special eigenvalues on functions and 1-forms. We compute and discuss the formulas in dimension n = 7.     

Determination of Dirac operator with eigenvalue-dependent boundary and jump conditions

Dirac operator with eigenvalue-dependent boundary and jump conditions is studied. Uniqueness theorems of the inverse problems from either Weyl function or the spectral data (the sets of eigenvalues and norming constants except for one eigenvalue and corresponding norming constant; two sets of different eigenvalues except for two eigenvalues) are proved. Finally, we investigate two applications of these theorems and obtain analogues of a theorem of Hochstadt-Lieberman and a theorem of Mochizuki-Trooshin.     

Inverse spectral problems for Dirac operator with eigenvalue dependent boundary and jump conditions

We deal with the Dirac operator with eigenvalue dependent boundary and jump conditions. Properties of eigenvalues, eigenfunctions and the resolvent operator are studied. Moreover, uniqueness theorems of the inverse problem according to the Weyl functions and the spectral data (the sets of eigenvalues and norming constants; two different eigenvalues sets) are proved.     

Estimates for weighted eigenvalues of a fourth-order elliptic operator with variable coefficients

We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in \mathbbRⁿ {\mathbb{R}^n} . We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the (k + 1)th eigenvalue in terms of the first k eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang–Xia inequality (J. Funct. Anal., 245, No. 1, 334–352 (2007)) for the clamped-plate problem to a fourth-order elliptic operator with variable coefficients.     

Eigenvalue estimate of the basic Dirac operator on a Kähler foliation

Let F be a Kähler spin foliation of codimension q=2n on a compact Riemannian manifold M with the transversally holomorphic mean curvature form κ. It is well known [S.D. Jung, T.H. Kang, Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation, J. Geom. Phys. 45 (2003) 75-90] that the eigenvalue λ of the basic Dirac operator Db satisfies the inequality , where σ∇ is the transversal scalar curvature of F. In this paper, we introduce the transversal Kählerian twistor operator and prove that the same inequality for the eigenvalue of the basic Dirac operator by using the transversal Kählerian twistor operator. We also study the limiting case. In fact, F is minimal and transversally Einsteinian of odd complex codimension n with nonnegative constant transversal scalar curvature.     

Extrinsic Bounds for Eigenvalues of the Dirac Operator

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spinor bundle of the normal bundle.     

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.     

Deforming the point spectra of one-dimensional Dirac operators

We provide a method of inserting and removing any finite number of prescribed eigenvalues into spectral gaps of a given one-dimensional Dirac operator. This is done in such a way that the original and deformed operators are unitarily equivalent when restricted to the complement of the subspace spanned by the newly inserted eigenvalue. Moreover, the unitary transformation operator which links the original operator to its deformed version is explicitly determined.

     

一类Dirac方程特征值问题的迹恒等式

讨论了带有非局部边界条件的一维Dirac方程BdY/dx+P(x)Y=λY的特征值问题,其中首先建立了问题的特征值集合与一个整函数u(λ)零点集合的对应,并对Dirac算子的特征值进行了估计,然后借助于一个积分恒等式,采用留数方法,得到了该问题的特征值的迹恒等式.     

ESTIMATES ON EIGENVALUES FOR THE BIHARMONIC OPERATOR ON A BOUNDED DOMAIN IN H~n（-1）

In this paper, we consider eigenvalues of the Dirichlet biharmonic operator on a bounded domain in a hyperbolic space. We obtain universal bounds on the (k + 1)th eigenvalue in terms of the first kth eigenvalues independent of the domains.     

Upper eigenvalue estimates for Dirac operators

We derive upper eigenvalue estimates for generalized Dirac operators on closed Riemannian manifolds. In the case of the classical Dirac operator the estimates on the first eigenvalues are sharp for spheres of constant curvature.     

Integral Equations and Operator Theory

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C₀ contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of Aⁿ, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A^* has trivial kernel.     

Inequalities for eigenvalues of the polyharmonic operator Δ p

In this paper we consider the bounds of the eigenvalues for a class of polyharmonic operators and obtain the bounds for (n+1)th eigenvalue interm of the firstn eigenvalues. Those estimates do not depend on the domain in which the problem is considered.     

Variational Principles for Eigenvalues of Self-adjoint Operator Functions

Variational principles for eigenvalues of certain functions whose values are possibly unbounded self-adjoint operators T() are proved. A generalised Rayleigh functional is used that assigns to a vector x a zero of the function T()x, x), where it is assumed that there exists at most one zero. Since there need not exist a zero for all x, an index shift may occur. Using this variational principle, eigenvalues of linear and quadratic polynomials and eigenvalues of block operator matrices in a gap of the essential spectrum are characterised. Moreover, applications are given to an elliptic eigenvalue problem with degenerate weight, Dirac operators, strings in a medium with a viscous friction, and a Sturm-Liouville problem that is rational in the eigenvalue parameter.     

Eigenvalue Estimates for Non-Selfadjoint Dirac Operators on the Real Line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with complex mass and non-Hermitian potential V lie in the disjoint union of two disks, provided that the L ¹-norm of V is bounded from above by the speed of light times the reduced Planck constant. The result is sharp; moreover, the analogous sharp result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of non-real eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.     

Eigenvalue estimates for Dirac operators with parallel characteristic torsion

Assume that the compact Riemannian spin manifold (Mn,g) admits a G-structure with characteristic connection ∇ and parallel characteristic torsion (∇T=0), and consider the Dirac operator D1/3 corresponding to the torsion T/3. This operator plays an eminent role in the investigation of such manifolds and includes as special cases Kostant's “cubic Dirac operator” and the Dolbeault operator. In this article, we describe a general method of computation for lower bounds of the eigenvalues of D1/3 by a clever deformation of the spinorial connection. In order to get explicit bounds, each geometric structure needs to be investigated separately; we do this in full generality in dimension 4 and for Sasaki manifolds in dimension 5.     

One‐dimensional approximations of the eigenvalue problem of curved rods

In this work we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of the linearized elasticity eigenvalue problem of curved rod‐like bodies with respect to the small thickness ? of the rod. We show that the eigenfunctions and scaled eigenvalues converge, as ? tends to zero, toward eigenpairs of the eigenvalue problem associated to the one‐dimensional curved rod model which is posed on the middle curve of the rod. Because of the auxiliary function appearing in the model, describing the rotation angle of the cross‐sections, the limit eigenvalue problem is non‐classical. This problem is transformed into a classical eigenvalue problem with eigenfunctions being inextensible displacements, but the corresponding linear operator is not a differential operator. Copyright © 2001 John Wiley & Sons, Ltd.     

Resonances of an elastic plate in a compressible confined fluid

We present a theoretical study of the resonances of a fluid–structureproblem, an elastic plate placed in a duct in the presence ofa compressible fluid. The case of a rigid plate has been largelystudied. Acoustic resonances are then associated to resonantmodes trapped by the plate. Due to the elasticity of the plate,we need to solve a quadratic eigenvalue problem in which theresonance frequencies k solve the equations (k) = k², where are the eigenvalues of a self-adjoint operator of the formA + kB. First, we show how to study the eigenvalues locatedbelow the essential spectrum by using the min–max principle.Then, we study the fixed-point equations. We establish sufficientconditions on the characteristics of the plate and of the fluidto ensure the existence of resonances. Such conditions are validatednumerically.