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.     

Spectrum of a model three-particle Schrödinger operator

We study the spectrum of a model three-particle Schrödinger operator H(ε), ε > 0. We prove that for a sufficiently small ε > 0, this operator has no bound states and no two-particle branches of the spectrum. We also obtain an estimate for the small parameter ε.     

Trace formulas for a discrete Schrödinger operator

The Schrödinger operator with complex decaying potential on a lattice is considered. Trace formulas are derived on the basis of classical results of complex analysis. These formulas are applied to obtain global estimates of all zeros of the Fredholm determinant in terms of the potential.     

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.     

On finiteness of discrete spectrum of three-particle Schrödinger operator on a lattice

On three-dimensional lattice we consider a system of three quantum particles (two of them are identical (fermions) and the third one is of another nature) that interact with the help of paired short-range gravitational potentials. We prove the finiteness of a number of bound states of respective Schrödinger operator in a case, when potentials satisfy some conditions and zero is a regular point for two-particle sub-Hamiltonian. We find a set of values for particles masses values such that Schrödinger operator may have only finite number of eigenvalues lying to the left of essential spectrum.     

A discrete Schrödinger operator on a graph

We consider the discrete Schrödinger operator on the graph obtained in the strong-coupling approximation from the standard electron Schrödinger operator in the system composed of a quantum wire and quantum dot. We investigate the general spectral properties of this operator and the problem of the existence and behavior of the eigenvalues and resonances depending on the small coupling constant. We study the scattering problem for weak potentials in the stationary approach.     

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.     

The number of eigenvalues of the two-particle discrete Schrödinger operator

We consider the two-particle discrete Schrödinger operator associated with the Hamiltonian of a system of two particles (fermions) interacting only at the nearest neighbor sites. We find the number and the location of the eigenvalues of this operator depending on the particle interaction energy, the system quasimomentum, and the dimension of the lattice ? ^ν , ν ≥ 1.     

The Schrödinger operator with a weakly accelerating potential

The ideas of scattering theory are applied to the construction of a unitary operator realizing the similarity of the operator - id/d in L₂() with a one-dimensional Schrödinger operator on the semiaxis with potential v(x), admitting at infinity the estimate.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 10–12, 1985.     

The first eigenvalue for a quasilinear Schrödinger operator and its application

In this paper, we study the first eigenvalue for a quasilinear Schrödinger operator, which is greater than the first eigenvalue of the usual laplacian operator. As an application we treat a quasilinear resonance problem involving a subcritical growth perturbation.     

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.     

An inverse spectral problem for a fractional Schrödinger operator

Archiv der Mathematik - We establish that the potential appearing in a fractional Schrödinger operator is uniquely determined by an internal spectral data.     

Asymptotic behavior of eigenvalues of the two-particle discrete Schrödinger operator

We consider two-particle Schrödinger operator H(k) on a three-dimensional lattice ? ³ (here k is the total quasimomentum of a two-particle system, $k \in \mathbb{T}^3 : = \left( { - \pi ,\pi ]^3 } \right)$ . We show that for any $k \in S = \mathbb{T}^3 \backslash ( - \pi ,\pi )^3$ , there is a potential $\hat v$ such that the two-particle operator H(k) has infinitely many eigenvalues z_n(k) accumulating near the left boundary m(k) of the continuous spectrum. We describe classes of potentials W(j) and W(ij) and manifolds S(j) ? S, i, j ∈ {1, 2, 3}, such that if k ∈ S(3), (k ₂ , k ₃ ) ∈ (?π,π) ² , and $\hat v \in W(3)$ , then the operator H(k) has infinitely many eigenvalues z_n(k) with an asymptotic exponential form as n → ∞ and if k ∈ S(i) ∩ S(j) and $\hat v \in W(ij)$ , then the eigenvalues z_nm(k) of H(k) can be calculated exactly. In both cases, we present the explicit form of the eigenfunctions.     

Schrödinger operator with a perturbed small steplike potential

We consider the Schrödinger operator with a potential that is periodic with respect to two variables and has the shape of a small step perturbed by a function decreasing with respect to a third variable. We show that under certain conditions on the magnitudes of the step and the perturbation, a unique level that can be an eigenvalue or a resonance exists near the essential spectrum. We find the asymptotic value of this level.     

A note on a discrete model for the harmonic oscillator Schrödinger equation in N-cartesian coordinates

We construct a discrete model for the time-independent harmonic oscillator Schrödinger partial differential equation and demonstrate that it can be separated into N ordinary difference equations for the case of N-cartesian space coordinates.     

The inverse problem for a nonlinear Schrödinger equation

Using variational methods, the solution of the inverse problem of finding the refractive index of a nonlinear medium in a multidimensional Schrödinger equation is studied. The correctness of the statement of the problem under consideration is investigated, and a necessary condition that must be satisfied by the solution of this problem is found. Bibliography: 8 titles.     

Existence of breathers for discrete nonlinear Schrödinger equations

In this paper, we obtain a new sufficient condition on the existence of breathers for the discrete nonlinear Schrödinger equations by using critical point theory in combination with periodic approximations. The classical Ambrosetti–Rabinowitz superlinear condition is improved.