Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice

We consider a system of three arbitrary quantum particles on a three-dimensional lattice that interact via short-range attractive potentials. We obtain a formula for the number of eigenvalues in an arbitrary interval outside the essential spectrum of the three-particle discrete Schrödinger operator and find a sufficient condition for the discrete spectrum to be finite. We give an example of an application of our results.     

About the Spectral Properties of One Three-Partial Model Operator

We investigate the structure of the essential spectrum of one three particle model operator H. We prove the existence of negative eigenvalues of the operator H and obtain the estimate for a number of negative eigenvalues of the operator H.     

On the number of eigenvalues of a matrix operator

We consider a matrix operator H in the Fock space. We prove the finiteness of the number of negative eigenvalues of H if the corresponding generalized Friedrichs model has the zero eigenvalue (0 = min σ _ess(H)). We also prove that H has infinitely many negative eigenvalues accumulating near zero (the Efimov effect) if the generalized Friedrichs model has zero energy resonance. We obtain asymptotics for the number of negative eigenvalues of H below z as z → −0.     

On infinity of the discrete spectrum of operators in the Friedrichs model

The discrete spectrumof selfadjoint operators in the Friedrichs model is studied. Necessary and sufficient conditions of existence of infinitely many eigenvalues in the Friedrichs model are presented. A discrete spectrum of a model three-particle discrete Schrödinger operator is described.     

The Efimov effect for a model “three-particle” discrete Schrödinger operator

We study the existence of an infinite number of eigenvalues for a model “three-particle” Schrödinger operator H. We prove a theorem on the necessary and sufficient conditions for the existence of an infinite number of eigenvalues of the model operator H below the lower boundary of its essential spectrum.     

Asymptotics of the discrete spectrum of a model operator associated with a system of three particles on a lattice

We consider a model Schrödinger operator H_μ associated with a system of three particles on the threedimensional lattice ? ³ with a functional parameter of special form. We prove that if the corresponding Friedrichs model has a zero-energy resonance, then the operator H_μ has infinitely many negative eigenvalues accumulating at zero (the Efimov effect). We obtain the asymptotic expression for the number of eigenvalues of H_μ below z as z → ?0.     

Spectrum of the One-dimensional Schrödinger Operator With a Periodic Potential Subjected to a Local Dilative Perturbation

We study the spectrum of the one-dimensional Schr?dinger operator with a potential, whose periodicity is violated via a local dilation. We obtain conditions under which this violation preserves the essential spectrum of the Schr?dinger operator and an infinite number of isolated eigenvalues appear in a gap of the essential spectrum. We show that the considered perturbation of the periodic potential is not relative compact in general.     

Asymptotics of the Distribution of Eigenvalues of a Selfadjoint Second Order Hyperbolic Differential Operator on the Two-Dimensional Torus

We study the asymptotic properties of the discrete spectrum for general selfadjoint second order hyperbolic operators on the two-dimensional torus. For a broad class of operators with sufficiently smooth coefficients and the principal part coinciding with the wave operator in the light cone coordinates we prove the discreteness of the spectrum and obtain an asymptotic formula for the distribution of eigenvalues. In some cases we can indicate the first two asymptotic terms. We discuss the relations of these questions to analytic number theory and mathematical physics.     

Essential and discrete spectra of the three-particle Schrödinger operator on a lattice

We study the position of the essential spectrum of a three-body Schrödinger operator H. We evaluate the lower boundary of the essential spectrum of H and prove that the number of eigenvalues located below the lower edge of the essential spectrum in the H model is finite.     

Spectral analysis of pseudodifferential operators

The subject of this paper is the spectral analysis of pseudodifferential operators in the framework of perturbation theory. We build up a closed extension (the closure, or the Friedrichs extension) of the perturbed operator. We also prove Weyl-type theorems on the invariance of the essential spectrum of the unperturbed operator. In the case when the perturbed operator is symmetric we obtain a self-adjoint extension. Finally, we consider the case of the relativistic, spin-zero Hamiltonian, with a large class of interactions containing both local potentials, like the Coulomb and Yukawa, and nonlocal ones.     

Variational principles for symmetric bilinear forms

Every compact symmetric bilinear form B on a complex Hilbert space produces, via an antilinear representing operator, a real spectrum consisting of a sequence decreasing to zero. We show that the most natural analog of Courant's minimax principle for B detects only the evenly indexed eigenvalues in this spectrum. We explain this phenomenon, analyze the extremal objects, and apply this general framework to the Friedrichs operator of a planar domain and to Toeplitz operators and their compressions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efimov's Effect in a Model of Perturbation Theory of the Essential Spectrum

A model operator similar to the energy operator of a system with a nonconserved number of particles is studied. The essential spectrum of the operator is described, and under some natural conditions on the parameters it is shown that there are infinitely many eigenvalues lying below the bottom of the essential spectrum.     

Embedded eigenvalues and resonances of a generalized Friedrichs model

The existence of resonances and embedded eigenvalues of a multidimensional generalized Friedrichs model is studied. The existence of a Friedrichs model with a given number of eigenvalues located within the continuous spectrum is proved. The existence of resonances is shown, and the widths of these resonances are calculated.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 1, pp. 54–62, April, 1995.     

On the Friedrichs extension of some block operator matrices

We give a matrix representation for the resolvent of the Friedrichs extension of some semibounded 2×2 operator matrices and study their essential spectrum.     

Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation

We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem.     

Spectral properties of the generalized Friedrichs model

We consider the self-adjoint generalized Friedrichs model with small values of the “coupling parameter.” In this case, we completely investigate the spectrum of the model and the structure of its eigenvectors (both ordinary and generalized). The constructions we use are based on an analysis of the resolvent of the Friedrichs operator and on the corresponding scattering theory.     

Investigation of the spectrum of a model operator in a Fock space

We consider a model operator H corresponding to a quantum system with a nonconserved finite number of particles on a lattice. Based on an analysis of the spectrum of the channel operators, we describe the position of the essential spectrum of H. We obtain a Faddeev-type equation for the eigenvectors of H.     

Some spectral properties of the generalized Friedrichs model

The spectrum and the resonance of the generalized Friedrichs model are studied. The existence of the wave operators and of the scattering operator is established. The structure of the resonances, the connection between resonances and eigenvalues, and the singularities of the scattering matrix are investigated.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 210–238, 1986.     

On the Finiteness of the Number of Eigenvalues of Jacobi Operators below the Essential Spectrum

We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As an application we show that Kneser's criterion for Jacobi operators follows as a special case.     

Global solutions of nonlinear Schrödinger equations

We study the nonlinear Schrödinger equation in \(\mathbb {R}^n\) without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in \([0, \infty )\) being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.