Three-dimensional Schrodinger Operator Research Articles

Spectral theory of Schrödinger operators with infinitely many point interactions and radial positive definite functions

A number of results on radial positive definite functions on Rn related to Schoenbergʼs integral representation theorem are obtained. They are applied to the study of spectral properties of self-adjoint realizations of two- and three-dimensional Schrödinger operators with countably many point interactions. In particular, we find conditions on the configuration of point interactions such that any self-adjoint realization has purely absolutely continuous non-negative spectrum. We also apply some results on Schrödinger operators to obtain new results on completely monotone functions.

Asymptotics of eigenelements of boundary value problems for the Schrödinger operator with a large potential localized on a small set

Asymptotics of eigenelements of a singularly perturbed boundary value problem for the three-dimensional Schrodinger operator is constructed in a bounded domain with the Dirichlet and Neumann boundary condition. The perturbation is described by a large potential whose support contracts into a point. In the case of the Dirichlet boundary conditions, this problem corresponds to a potential well with infinitely high walls and a narrow finite peak at the bottom.

Spectral shift function in strong magnetic fields

The three-dimensional Schrodinger operator H with constant magnetic field of strength b> 0 is considered under the assumption that the electric potential V ∈ L 1 (R 3 ) admits certain power-like estimates at infinity. The asymptotic behavior as b →∞ of the spectral shift function ξ(E; H, H0) is studied for the pair of operators (H, H0 )a t the energiesE = Eb + λ, E > 0a ndλ ∈ R being fi xed. Two asymptotic regimes are distinguished. In the first regime, called asymptotics far from the Landau levels ,w e pickE/2 � Z+ and λ ∈ R; then the main term is always of order √ b ,a nd is independent of λ. In the second asymptotic regime, called asymptotics near a Landau level ,w e chooseE =2 q0, q0 ∈ Z+ ,a ndλ � 0; in this case the leading term of the SSF could be of order b or √ b for different λ.

On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

We consider the three-dimensional Schrodinger operators \( H_0 \) and \( H_\pm \) where \( H_{0} = (i\nabla + A)^{2} - b \) , A is a magnetic potential generating a constant magnetic field of strength \( b > 0 \) , and \( H_{\pm} = H_{0} \pm V \) where \( b \geq 0 \) decays fast enough at infinity. Then, A. Pushnitski’s representation of the spectral shift function (SSF) for the pair of operators \( H_\pm, H_0 \) is well defined for energies \( E \neq 2qb,\, q \in \mathbb{Z}_{+}. \) We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels \( 2qb,\, q \in \mathbb{Z}_{+}. \) Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity.

Absolutely continuous spectrum of multidimensional Schrödinger operator

We prove that a three-dimensional Schrodinger operator with slowly decaying potential has an absolutely continuous (a.c.) spectrum that fills R+. Asymptotics of Green's functions is obtained as well.

New Approach to Quantum Scattering Near the Lowest Landau Threshold for a Schrödinger Operator with a Constant Magnetic Field

For a fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrodinger operator with a constant magnetic field and an axisymmetrical electric potential V. Asymptotic expansions for the resolvent of the Hamiltonian H m = H om + V are deduced as the spectral parameter tends to the lowest Landau threshold E0. In particular it is shown that E0 can be an eigenvalue of H m . Furthermore, asymptotic expansions of the scattering matrix associated with the pair (H m , H om ) are derived as the energy parameter tends to E0.

Strong Magnetic Field Asymptotics of Integrated Density of States for a Random 3D Schrödinger Operator

We consider the three-dimensional Schrodinger operator with constant magnetic field and bounded random electric potential. We investigate the asymptotic behaviour of the integrated density of states for this operator as the norm of the magnetic field tends to infinity.

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 Schrodinger 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.

Spectrum of three-dimensional landau operator perturbed by a periodic point potential

A study is made of a three-dimensional Schrodinger operator with magnetic field and perturbed by a periodic sum of zero-range potentials. In the case of a rational flux, the explicit form of the decomposition of the resolvent of this operator with respect to the spectrum of irreducible representations of the group of magnetic translations is found. In the case of integer flux, the explicit form of the dispersion laws is found, the spectrum is described, and a qualitative investigation of it is made (in particular, it is established that not more than one gap exists).

THE MULTIDIMENSIONAL SCHRÖDINGER OPERATOR WITH A PERIODIC POTENTIAL

In this paper the author investigates the zonal structure of the spectrum of the three-dimensional Schrodinger operator with periodic potential. The main result is an estimate of the number n(λ) of zones of the spectrum covering the real point λ. It is shown that, under certain conditions on the period lattice of the potential, n(λ)>λ when λ→∞. From this estimate it follows that the number of lacunae in the spectrum of the Schrodinger operator is finite. It is also shown that for periodic potentials with small norm there are in general no lacunae in the spectrum. Analogous results are formulated for the Schrodinger operator in higher dimensions. Bibliography: 17 titles.

The spectrum of defect periodic point interactions

We show that the negative part of the spectrum of the three-dimensional Schrodinger operator with periodic point interactions with a finite number n of centers in each periodic cell consists of, at most, n bands. Furthermore, we show that if one removes the interaction at some arbitrary points the negative part of the spectrum is contained within the lower and upper edge of the negative part of the spectrum of the original operator. More detailed properties of the spectrum in the case n=2 or 3 are also given.

The virtual level of the Schr�dinger equation

The three-dimensional Schrodinger operator H is considered with a so-called virtual level at the start of the spectrum. The existence of a virtual level means, roughly speaking, that the corresponding Lippman-Schwinger equation has a nontriviai, zero-energy solution. The concept of a virtual level is discussed in detail, and the structure of the resolvent of the operator H is studied.