The infiniteness of the number of eigenvalues in the gap in the essential spectrum for the three-particle Schrödinger operator on a lattice

We consider a system of three arbitrary quantum particles on a three-dimensional lattice that interact via attractive pair contact potentials. We find a condition for a gap to appear in the essential spectrum and prove that there are infinitely many eigenvalues of the Hamiltonian of the corresponding three-particle system in this gap. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 2, pp. 299–317, May, 2009.     

Spectrum of the three-particle Schr?dinger operator on a one-dimensional lattice

We consider a system of three arbitrary quantum particles on a one-dimensional lattice interacting pairwise via attractive contact potentials. We prove that the discrete spectrum of the corresponding Schr?dinger operator is finite for all values of the total quasimomentum in the case where the masses of two particles are finite. We show that the discrete spectrum of the Schr?dinger operator is infinite in the case where the masses of two particles in a three-particle system are infinite.     

Elastic state of a plate with a circular plug and a rectilinear thin elastic inclusion

There is considered the problem of the state of stress of an infinite elastic plane with a bonded circular plug and an arbitrarily located thin elastic inclusion under biaxial tension. Conditions of ideal mechanical contact are satisfied on the line separating the materials. By using the complex Kolosov — Muskhelishvili potentials, the problem is reduced to a system of integro-differential equations which is solved numerically by utilization of a mechanical quadrature method. A numerical analysis is given for the solution of the problem of the elastic equilibrium of a plane with a circular hole and an arbitrarily located thin inclusion.     

Solution of a coupled system of differential equations

We give the fundamental solution of the coupled system of differential equations of the dynamic thermoelasticity problem, use it to introduce thermoelastic and thermal simple-layer, double-layer, and volume retarded potentials, and analyze the properties of these potentials in domains with nonsmooth boundaries. We construct the solution of this system as well as of the corresponding transmission problem in domains with nonsmooth boundaries.     

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.     

On the solution of boundary value problems on an inhomogeneous plane with a crack and a barrier connected in series

We consider boundary value problems for an equation in divergence form on a plane divided into two inhomogeneous half-planes by a film inclusion in the form of a strongly permeable crack and a weakly permeable barrier connected in series; this models a contact of heterogeneous media under inhomogeneous external conditions. The desired potentials have prescribed singular points (sources, drains, etc.). The coefficients of the equation are nonconstant and may increase or decrease when moving away from the film inclusion along a family of parabolas. We obtain representations of solutions of the considered problems via harmonic functions with the corresponding singular points on the plane.     

Essential spectrum of a model operator associated with a three-particle system on a lattice

We consider a model operator H associated with the system of three particles interacting via nonlocal pair potentials on a ν-dimensional lattice. We identify channel operators and use their spectra to describe the position and structure of the essential spectrum of H. We obtain an analogue of the Faddeev equation for the eigenfunctions of H.     

Problem of thermoelectromagnetoelasticity for a piezoelectric/piezomagnetic bimaterial with interface crack

We consider a plane strain problem for a piezoelectric/piezomagnetic bimaterial space with a crack in the region of the interface of the materials. At infinity, tensile and shear stresses and heat, electric, and magnetic flows are set. Using representations for all mechanical, thermal, and electromagnetic factors in terms of piecewise analytic functions, we formulate problems of linear conjugation that correspond to a model of an open crack and models taking into account the contact zone in the vicinity of a crack tip. Exact analytic solutions of the indicated problems are constructed. Expressions for stresses, the electric and magnetic inductions, jumps of derivatives of displacements, and electric and magnetic potentials on the interface are written. The coefficients of intensities of the indicated factors are presented. We derive a transcendental equation for the determination of the real length of the contact zone. The dependences of this length and the coefficients of intensity on the set external influences are investigated.     

On a thermal stress problem for contacting half-spaces with inclusions of other material involving a new method for computing potentials and singular integrals

This paper deals with a thermal stress problem for contacting half-spaces of different materials having an inclusion of other material. With the aid of the contact tensor of two half spaces the problem is reduced to singular integral equations. Some new theorems on potentials with the contact tensor are used. Finally, a potential of the double layer with the contact tensor in the kernel and a singular integral operator, being the direct value of the potential, are calculated. For this a Kupradze method is applied, interpreting this potential as a solution of a contact problem of elastostatics. Some numerical experiments are communicated.     

Differential variational–hemivariational inequalities with application to contact mechanics

In this paper we study a dynamical system which consists of the Cauchy problem for a nonlinear evolution equation of first order coupled with a nonlinear time-dependent variational–hemivariational inequality with constraint in Banach spaces. The evolution equation is considered in the framework of evolution triple of spaces, and the inequality which involves both the convex and nonconvex potentials. We prove existence of solution by the Kakutani–Ky Fan fixed point theorem combined with the Minty formulation and the theory of hemivariational inequalities. We illustrate our findings by examining a nonlinear quasistatic elastic frictional contact problem for which we provide a result on existence of weak solution.     

Finiteness of the discrete spectrum of the Schrödinger operator of three particles on a lattice

We consider a system of three quantum particles interacting by pairwise short-range attraction potentials on a three-dimensional lattice (one of the particles has an infinite mass). We prove that the number of bound states of the corresponding Schrödinger operator is finite in the case where the potentials satisfy certain conditions, the two two-particle sub-Hamiltonians with infinite mass have a resonance at zero, and zero is a regular point for the two-particle sub-Hamiltonian with finite mass.     

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

We consider a model operator H associated with a system of three particles on a lattice interacting via nonlocal pair potentials. Under some natural conditions on the parameters specifying this model operator H, we prove the finiteness of its discrete spectrum.     

Regularity of solutions to a contact problem

We consider a variational inequality for the Lamé system which models an elastic body in contact with a rigid foundation. We give conditions on the domain and the contact set which allow us to prove regularity of solutions to the variational inequality. In particular, we show that the gradient of the solution is a square integrable function on the boundary.

     

Perturbation theory in the scattering problem for a three-particle system

We consider the scattering problem for a system of three nonrelativistic particles in the case of energies below the threshold of the system breakup into three free particles. We assume that the interaction potentials can be represented as a sum of two terms, one of which is a small perturbation. We develop a perturbation theory scheme for solving the scattering problem based on the three-particle Faddeev equations.     

Steady-State Frictional Heat Generation on a Periodic Sliding Contact

We investigate the motion of a periodic system of rigid, isolated, parabolic dies along the surface of a half-space. The action of friction forces results in heat generation in the contact region. We assume that the surfaces of the dies are thermally insulated and the half-space is a heat conductor. We reduce the problem under consideration to a set of two integral equations for the contact temperature and pressure. We solve these equations numerically and investigate the influence of thermal deformation on the distributions of temperature and pressure and on the dimensions of the contact region.     

Finiteness of the Discrete Spectrum of the Hamiltonian of a System of Three Arbitrary Particles on a Lattice

We prove that the number of bound states for the Hamiltonian of a system of three arbitrary particles interacting through pairwise attraction potentials on a three-dimensional lattice is finite in the cases where (1) none of the two-particle subsystems has a virtual level and (2) only one of the two-particle subsystems has a virtual level.     

Numerical solutions of transonic two-dimensional flows at a ninety-degree wedge

We present numerical results on self-similar two-dimensional Riemann problems governed by the compressible Euler system and the nonlinear wave system, which give rise to a transonic shock. We consider a configuration for a vertical incident shock moving to the right above a rectangular object. The incident shock then interacts with a sonic circle soon after it moves beyond the object, and creates a transonic region. We implement Lax–Liu positive schemes and Strang splitting, and obtain linear correlations of the incident shock strength and the shock strength at the vertical wall. We further implement Roe average methods and finite volume methods on quadrilateral grids to capture a contact discontinuity of the Euler system near the corner of the object. The contact discontinuity creates a new supersonic state and a transonic shock inside the transonic region.     

Capillary oscillations at a circular orifice

We compute the normal frequencies and normal modes for the oscillation of the free surface of a perfect incompressible fluid inside a semi-infinite container with a circular orifice. In doing that, a dual integral equation system involving the Bessel functions must be solved. We discuss the cases where the contact line between the free surface and the container is pinned as well as the case where it moves with a constant contact angle.     

The stress state of a plastic layer with a variable yield strength under a flat deformation

We study the stress state of a plastic layer with a variable yield strength in a strip under a flat deformation with a tensile load. We approximately calculate the first integrals of the system of plastic equilibrium equations, obtain an analog of the first Hencky theorem, and solve the conjugation problem for stresses on the contact boundary.