Branches of the essential spectrum of the lattice spin-boson model with at most two photons

We consider a lattice analogue of the A_m model of light radiation with a fixed atom and at most m photons (m = 1, 2). We describe the essential spectrum of the operator A₂ in terms of the spectrum of the operator A₁, i.e., we find the “two-particle” and “three-particle” branches of the essential spectrum of A₂. We prove that the essential spectrum is a union of at most six intervals, and we study their positions. We derive an estimate for the lower bound of the “two-particle” and “three-particle” branches.     

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.     

Finiteness of the discrete spectrum of the three-particle schrödinger equation on a lattice

Consideration is given to the Hamiltonian of a system of three identical quantum particles on a lattice that interact via pairwise contact attractive potentials. Finiteness of the three-particle bound states is proved for the three-dimensional discrete Schrödinger operator on the condition that the operators describing the two-particle subsystems have no virtual levels. For high dimensions (v ≥ 5), the finiteness of three-particle bound states is also proved in the presence of virtual levels.     

Faddeev differential equations as a spectral problem for a nonsymmetric operator

We consider a nonsymmetric matrix operator whose eigenvalue problem is the system of Faddeev differential equations for a three-particle system. For this operator and its adjoint, the resolvents are represented in terms of Faddeev T-matrix components of the three-particle Schrödinger operator. On the basis of these representations, the invariant spaces of the operators under consideration are investigated and their eigenfunctions are determined. The biorthogonality and completeness of the eigenfunction system are proved.We dedicate this paper to the memory of Stanislav Petrovitch Merkuriev, who left us three years ago.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 513–528, June, 1996.     

Spectrum of the Three-Particle Schrödinger Difference Operator on a Lattice

For a three-particle operator on a lattice, we study the properties of its spectrum that depend on pairwise interactions and are determined by a parameter characterizing the intensity of interaction.     

On the spectrum of a vector Schrödinger operator

We consider the asymptotics of the spectrum of a Sturm-Liouville operator acting on a space of vector functions and show that this asymptotics is affected by “rotation” of eigenvectors of the potential. A similar result is obtained for a vector Schrödinger operator.     

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 Recursion Operator of the Kadomtsev-Petviashvili Equation and the Squared Eigenfunctions of the Schrödinger Operator

The recursion operator of the Kadomtsev-Petviashvili equation is algorithmically derived. This recursion operator is the two-spatial-dimensional analogue of the Lenard operator of the Korteweg-deVries equation. It is also the “squared” eigenfunction operator of the time-dependent Schrödinger operator. The existence of the recursion operator suggests that the Kadomtsev-Petviashvili equation is a hi-Hamiltonian system.     

A discrete “three-particle” Schrödinger operator in the Hubbard model

In the space L ₂(T ^ν ×T ^ν ), where T ^ν is a ν-dimensional torus, we study the spectral properties of the “three-particle” discrete Schrödinger operator ? = H₀ + H₁ + H₂, where H₀ is the operator of multiplication by a function and H₁ and H₂ are partial integral operators. We prove several theorems concerning the essential spectrum of ?. We study the discrete and essential spectra of the Hamiltonians H^t and h arising in the Hubbard model on the three-dimensional lattice.     

A maximum of the first eigenvalue of semibounded differential operator with a parameter

We consider a self-adjoint differential operator in Hilbert space. Then the domain of the operator is changed by the perturbation of the boundary conditions so that a given neighborhood “is cleared” from the points of the spectrum of the perturbed operator. For the Sturm–Liouville operator on the segment and the Laplace operator on the square such a possibility is attained via integral perturbations of boundary conditions.     

Remarks on the spectral theory for the multiparticle-type Schrödinger operator

Mourre's method is used to prove the limiting absorption principle for the multiparticle Schrödinger operator under the same assumptions on the pair potentials as in the two-particle problem. It is shown that at high energies this principle is valid under wider conditions than on the whole spectral axis. The scattering theory for a Schrödinger operator whose potential decays at infinity in an essentially anisotropic manner is constructed in analogy with the three-particle problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 277–298, 1984.     

Spectral properties of a Hamiltonian of a four-particle system on a lattice

We consider the Hamiltonian of a system of four arbitrary quantum particles with two-particle contact (noncompact) interaction potentials on a three-dimensional lattice perturbed by three-particle contact potentials. We describe the location of the essential spectrum of the Schrödinger operator corresponding to a four-particle system.     

S-Storage Operators

In 1990, J. L. Krivine introduced the notion of storage operator to simulate, for Church integers, the “call by value” in a context of a “call by name” strategy. In the present paper we define for every λ-term S which realizes the successor function on Church integers the notion of S-storage operator. We prove that every storage operator is an S-storage operator. But the converse is not always true.     

关于赵君喜“R_n~∞中一个幂等元的构造”一文的意见

本文证明了“中一个幂等元的构造”一文中的幂等元Ｑ不是套代数ａｌｇＮ中的算子，因此也不是如该文作者所断言的是中的元。     

Contraction mappings underlying undiscounted Markov decision problems

This paper is concerned with the properties of the value-iteration operator0 which arises in undiscounted Markov decision problems. We give both necessary and sufficient conditions for this operator to reduce to a contraction operator, in which case it is easy to show that the value-iteration method exhibits a uniform geometric convergence rate. As necessary conditions we obtain a number of important characterizations of the chain and periodicity structures of the problem, and as sufficient conditions, we give a general “scrambling-type” recurrency condition, which encompasses a number of important special cases. Next, we show that a data transformation turns every unichained undiscounted Markov Renewal Program into an equivalent undiscounted Markov decision problem, in which the value-iteration operator is contracting, because it satisfies this “scrambling-type” condition. We exploit this contraction property in order to obtain lower and upper bounds as well as variational characterizations for the fixed point of the optimality equation and a test for eliminating suboptimal actions.     

On the quasi-stationary Maxwell equations with monotone characteristics in a multiply connected domain

The quasi-stationary Maxwell equations for a magnetic core with a non-linear characteristic and homeomorphic to a three dimensional torus are reduced to a unique evolution equation involving a monotone operator. This reduction is based on the properties of some functional spaces suited to the study of the operators “curl” and “div,” with different kinds of boundary and period conditions. Some existence, uniqueness and regularity results are proved.     

On the asymptotics of the spectrum of a nonsemibounded vector Sturm-Liouville operator

On the half-line, we consider a vector Sturm-Liouville operator with a potential that is unbounded below. Asymptotic formulas for the spectrum are given. These formulas involve the eigenvalues of the matrix potential as well as the “rotational velocities” of the eigenvectors.     

Pointwise convergence to the initial data for nonlocal dyadic diffusions

We solve the initial value problem for the diffusion induced by dyadic fractional derivative s in ?⁺. First we obtain the spectral analysis of the dyadic fractional derivative operator in terms of the Haar system, which unveils a structure for the underlying “heat kernel”. We show that this kernel admits an integrable and decreasing majorant that involves the dyadic distance. This allows us to provide an estimate of the maximal operator of the diffusion by the Hardy-Littlewood dyadic maximal operator. As a consequence we obtain the pointwise convergence to the initial data.     

Carleman Parabola and the Eigenvalues of Elliptic Operators

We study the problem of location of the spectrum of an elliptic operator in a bounded domain with both the classical and more special boundary conditions. On the complex plane, we construct the set of all eigenvalues of the corresponding operator and consider several examples to illustrate the “asymptotic accuracy” of the obtained set and consider an example of nonunique solvability of a parabolic inverse problem with final observation.     

基于直觉模糊集的世界一流学科建设成效评价

针对目前我国世界一流学科建设成效评价存在主客观偏好,建立了直觉模糊综合评判法的数学模型。基于直觉模糊集,提出了两种直觉模糊合成算子,即直觉模糊加权平均算子(IFWA)和直觉模糊主因素突出算子(IFPFP),将其应用于世界一流学科建设成效的评价。应用实例表明,该方法有效地解决了目前学科评价领域中部分评价指标量化困难、代表性差等突出问题,可给出更加科学、准确的评价结果。