Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit

A perturbation decaying to at and not too irregular at introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues in a compact subset of a gap in the essential spectrum is given by a quasi-semiclassical asymptotic formula in the slow-decay limit, which for power-decaying perturbations is equivalent to the large-coupling limit. This asymptotic behaviour elucidates the origin of the dense point spectrum observed in spherically symmetric, radially periodic three-dimensional Dirac operators.

     

Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics

The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice and interacting via zero-range attractive potentials is considered. For the two-particle energy operator h(k), with the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of h(k) for k 0 is proven, provided that h(0) has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number N(0, z) of eigenvalues of H(0) lying below the following limit exists with Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum K the finiteness of the number of eigenvalues of H(K) below the essential spectrum is established and the asymptotics for the number N(K, 0) of eigenvalues lying below zero is given. Communicated by Gian Michele GrafSubmitted 19/11/03, accepted 08/03/04     

A finite-difference approximation for the eigenvalues of the clamped plate

Summary A finite-difference scheme is given for the eigenproblem of the clamped plate. The discrete eigenvalues and eigenvectors are shown to converge to the continuous eigenvalues and eigenvectors likeO(h ²) andO (h ² logh 1/2) respectively.This work, supported by the U.S. Department of the Navy under Contract N 00017-62-C-0604.     

The Asymptotics of the Number of Eigenvalues of a Three-Particle Lattice Schrödinger Operator

The Hamiltonian of a system of three quantum-mechanical particles moving on the three-dimensional lattice and interacting via zero-range attractive potentials is considered. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), where K is the three-particle quasimomentum, is studied. The absence of eigenvalues below the bottom of the essential spectrum of H(K) for all sufficiently small values of the zero-range attractive potentials is established.The asymptotics is found for the number of eigenvalues N(0,z) lying below . Moreover, for all sufficiently small nonzero values of the three-particle quasimomentum K, the finiteness of the number of eigenvalues below the essential spectrum of H(K) is established and the asymptotics of the number N(K,0) of eigenvalues of H(K) below zero is given.     

Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices

The aim of this paper is to find asymptotic formulas for eigenvalues of self-adjoint discrete operators in given by some infinite symmetric Jacobi matrices. The approach used to calculate an asymptotic behaviour of eigenvalues is based on method of diagonalization, Janas and Naboko’s lemma [J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36(2) (2004) 643–658] and the Rozenbljum theorem [G.V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle, (Russian) Trudy Maskov. Mat. Obshch. 36 (1978) 59–84]. The asymptotic formulas are given with use of eigenvalues and determinants of finite tridiagonal matrices.     

Maaß cusp forms for large eigenvalues

We investigate the numerical computation of Maaß cusp forms for the modular group corresponding to large eigenvalues. We present Fourier coefficients of two cusp forms whose eigenvalues exceed . These eigenvalues are the largest that have so far been found in the case of the modular group. They are larger than the millionth eigenvalue.     

A quantum waveguide with Aharonov–Bohm magnetic field

In a previous study, we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width d. We impose the Neumann boundary condition on a disk window of radius a and Dirichlet boundary conditions on the remained part of the boundary of the strip. We proved that such system exhibits discrete eigenvalues below the essential spectrum for any a > 0. In the present work, we study the effect of the presence of a magnetic field of Aharonov–Bohm type on this system. Precisely, we prove that in the presence of such field, there is some critical values of a₀>0, for which we have absence of the discrete spectrum for . We give a sufficient condition for the existence of discrete eigenvalues. Copyright © 2015 John Wiley & Sons, Ltd.     

Small potential corrections for the discrete eigenvalues of the Sturm-Liouville problem

Summary The inverse Sturm-Liouville problem is the problem of finding a good approximation of a potential functionq such that the eigenvalue problem (^*)–y +qy=y holds on (0, ) fory(0)=y()=0 and a set of given eigenvalues . Since this problem has to be solved numerically by discretization and since the higher discrete eigenvalues strongly deviate from the corresponding Sturm-Liouville eigenvalues , asymptotic corrections for the 's serve to get better estimates forq. Let _k (1kn) be the first eigenvalues of (^*), let _k be the corresponding discrete eigenvalues obtained by the finite element method for (^*) and let _{k k} for the special caseq=0. Then, starting from an asymptotic correction technique proposed by Paine, de Hoog and Anderssen, new estimates for the errors of the corrected discrete eigenvalues are obtained and confirm and improve the knownO(kh ²)(h:=/(n+1)) behaviour. The estimates are based on new Sobolev inequalities and on Fourier analysis and it is shown that for 4+c ₂ k(n+1)/2, wherec ₁ andc ₂ are constants depending onq which tend to 0 for vanishingq.     

The approximation of the Maxwell eigenvalue problem using a least-squares method

In this paper we consider an approximation to the Maxwell's eigenvalue problem based on a very weak formulation of two div-curl systems with complementary boundary conditions. We formulate each of these div-curl systems as a general variational problem with different test and trial spaces, i.e., the solution space is and components in the test spaces are in subspaces of , the Sobolev space of order one on the computational domain . A finite-element least-squares approximation to these variational problems is used as a basis for the approximation. Using the structure of the continuous eigenvalue problem, a discrete approximation to the eigenvalues is set up involving only the approximation to either of the div-curl systems. We give some theorems that guarantee the convergence of the eigenvalues to those of the continuous problem without the occurrence of spurious values. Finally, some results of numerical experiments are given.

     

Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum

We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential of a one-dimensional Schrödinger operator determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of on a finite interval and knowledge of over a corresponding fraction of the interval. The methods employed rest on Weyl -function techniques and densities of zeros of a class of entire functions.

     

On the first two eigenvalues of Sturm-Liouville operators

Among the Schrödinger operators with single-well potentials defined on with transition point at , the gap between the first two eigenvalues of the Dirichlet problem is minimized when the potential is constant. This extends former results of Ashbaugh and Benguria with symmetric single-well potentials. An analogous result is given for the Dirichlet problem of vibrating strings with single-barrier densities for the ratio of the first two eigenvalues.

     

Distribution of Hecke eigenvalues

We give two results concerning the distribution of Hecke eigenvalues of . The first result asserts that on certain average the Sato-Tate conjecture holds. The second result deals with the Gaussian central limit theorem for Hecke eigenvalues.

     

Lower bounds for the energy of digraphs

The energy of a digraph D is defined as , where z₁,…,z_n are the eigenvalues of D. In this article we find lower bounds for the energy of digraphs in terms of the number of closed walks of length 2, extending in this way the result obtained by Caporossi et al. [G. Caporossi, D. Cvetkovi?, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996]: for all graphs G with m edges. Also, we study digraphs with three eigenvalues.     

Existence and asymptotic stability of relaxation discrete shock profiles

In this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws. The existence of discrete shocks is established by suitable manifold construction, and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in , provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the relaxing scheme for the single shock solution of the system of hyperbolic conservation laws. These results are proved by using both a weighted norm estimate and a characteristic energy method based on the internal structures of the discrete shocks.

     

Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpinski gasket

In this note we investigate the asymptotic behavior of spectra of Schrödinger operators with continuous potential on the Sierpinski gasket . In particular, using the existence of localized eigenfunctions for the Laplacian on we show that the eigenvalues of the Schrödinger operator break into clusters around certain eigenvalues of the Laplacian. Moreover, we prove that the characteristic measure of these clusters converges to a measure. Results similar to ours were first observed by A. Weinstein and V. Guillemin for Schrödinger operators on compact Riemannian manifolds.

     

Asymptotics of Plancherel measures for symmetric groups

We consider the asymptotics of the Plancherel measures on partitions of as goes to infinity. We prove that the local structure of a Plancherel typical partition in the middle of the limit shape converges to a determinantal point process with the discrete sine kernel.

On the edges of the limit shape, we prove that the joint distribution of suitably scaled 1st, 2nd, and so on rows of a Plancherel typical diagram converges to the corresponding distribution for eigenvalues of random Hermitian matrices (given by the Airy kernel). This proves a conjecture due to Baik, Deift, and Johansson by methods different from the Riemann-Hilbert techniques used in their original papers and from the combinatorial proof given by the second author. Our approach is based on an exact determinantal formula for the correlation functions of the poissonized Plancherel measures in terms of a new kernel involving Bessel functions. Our asymptotic analysis relies on the classical asymptotic formulas for the Bessel functions and depoissonization techniques.

     

On the existence of eigenvalues of Toeplitz operators on planar regions

A study is made of the eigenvalues of self-adjoint Toeplitz operators on multiply connected planar regions having holes. The presence of eigenvalues is detected through an analysis of the zeros of translations of theta functions restricted to in .