Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential

We consider the linear Schrödinger equation on a one dimensional torus and its time-discretization by splitting methods. Assuming a non-resonance condition on the stepsize and a small size of the potential, we show that the numerical dynamics can be reduced over exponentially long time to a collection of two dimensional symplectic systems for asymptotically large modes. For the numerical solution, this implies the long time conservation of the energies associated with the double eigenvalues of the free Schrödinger operator. The method is close to standard techniques used in finite dimensional perturbation theory, but extended here to infinite dimensional operators.     

Semiclassical asymptotic approximations and the density of states for the two-dimensional radially symmetric Schrödinger and Dirac equations in tunnel microscopy problems

We consider the two-dimensional stationary Schrödinger and Dirac equations in the case of radial symmetry. A radially symmetric potential simulates the tip of a scanning tunneling microscope. We construct semiclassical asymptotic forms for generalized eigenfunctions and study the local density of states that corresponds to the microscope measurements. We show that in the case of the Dirac equation, the tip distorts the measured density of states for all energies.     

Splitting of lower energy levels in a quantum double well in a magnetic field and tunneling of wave packets in Nanowires

We consider the problem of the splitting of lower eigenvalues of the two-dimensional Schrödinger operator with a double-well-type potential in the presence of a homogeneous magnetic field. The main result is the observation that the partial Fourier transformation takes the operator under study to a Schrödingertype operator with a (new) double-well-type potential but already without any magnetic field. We use this observation to investigate the influence of the magnetic field on the tunneling effects. We discuss two methods for calculating the splitting of lower eigenvalues: based on the instanton and based on the so-called libration. We use the obtained result to study the tunneling of wave packets in parallel quantum nanowires in a constant magnetic field.     

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.     

On the asymptotics of solutions to the second initial boundary value problem for Schrödinger systems in domains with conical points

In this paper, for the second initial boundary value problem for Schrödinger systems, we obtain a performance of generalized solutions in a neighborhood of conical points on the boundary of the base of infinite cylinders. The main result are asymptotic formulas for generalized solutions in case the associated spectrum problem has more than one eigenvalue in the strip considered.     

The spectrum of the Schrödinger operator with a rapidly oscillating compactly supported potential

We study the effect of an eigenvalue appearing from the boundary of the essential spectrum of the Schrödinger operator perturbed by a rapidly oscillating compactly supported potential. We prove sufficient conditions for the existence and absence of such an eigenvalue and obtain the first few terms of its asymptotic expansion for the case where this eigenvalue exists.     

NON-CLASSICAL ASYMPTOTICS OF THE TRACE OF THE HEAT KERNEL FOR THE MAGNETIC SCHRÖDINGER OPERATOR

We consider the effect of a magnetic field for the asymptotic behavior of the trace of the heat kernel for the Schrödinger operator. We discuss the case where the operator has compact resolvents in spite of the fact that the electric potential is degenerate on some submanifold. According to the degree of the degeneracy, we obtain non-classical asymptotics.     

Electron scattering by a crystal layer

We consider the one-particle discrete Schrödinger operator H with a periodic potential perturbed by a function ?W that is periodic in two variables and exponentially decreasing in the third variable. Here, ? is a small parameter. We study the scattering problem for H near the point of extremum with respect to the third quasimomentum coordinate for a certain eigenvalue of the Schrödinger operator with a periodic potential in the cell, in other words, for the small perpendicular component of the angle of particle incidence on the potential barrier ?W. We obtain simple formulas for the transmission and reflection probabilities.     

Quasilevels of a two-particle Schrödinger operator with a perturbed periodic potential

We consider a two-dimensional periodic Schrödinger operator perturbed by the interaction potential of two one-dimensional particles. We prove that quasilevels (i.e., eigenvalues or resonances) of the given operator exist for a fixed quasimomentum and a small perturbation near the band boundaries of the corresponding periodic operator. We study the asymptotic behavior of the quasilevels as the coupling constant goes to zero. We obtain a simple condition for a quasilevel to be an eigenvalue.     

On the range of use of the tight-binding approximation method for a complex-valued potential

We consider a one-dimensional Schrödinger operator with a periodic potential constructed as the sum of the shifts of a given complex-valued potential q that decreases at infinity. Mathematical justification of the tight-binding approximation method is presented. Let λ₀ be an isolated eigenvalue of the Schrödinger operator with a potential q. Then, for the corresponding operator with a periodic potential, a continuous spectrum exists in the neighborhood of λ₀. The asymptotic behavior of this part of the spectrum as the period increases infinitely is studied for the cases of one- and two-dimensional invariant subspaces corresponding to λ₀.     

Reduction of the dressing chain of the Schrödinger operator

We reduce the problem of constructing real finite-gap solutions of the focusing modified Korteweg-de Vries equation, to the dressing chain of the Schrödinger operator. We show that the Schrödinger operator spectral curve corresponding to such a solution is real. We give some restrictions on the initial data for the chain that lead to such solutions. We also consider a soliton, reduction. We obtain compact representations for the multisoliton and breather solutions of the modified Korteweg-de Vries equations; these representations can be useful in developing the perturbation theory for various applied problems.     

Stiff convergence of force-gradient operator splitting methods

We consider force-gradient, also called modified potential, operator splitting methods for problems with unbounded operators. We prove that force-gradient operator splitting schemes retain their classical orders of accuracy for linear time-dependent partial differential equations of parabolic and Schrödinger types, provided that the solution is sufficiently regular.     

Lower part of the spectrum for the two-dimensional Schrödinger operator periodic in one variable and application to quantum dimers

We study the semiclassical asymptotic approximation of the spectrum of the two-dimensional Schrödinger operator with a potential periodic in x and increasing at infinity in y. We show that the lower part of the spectrum has a band structure (where bands can overlap) and calculate their widths and dispersion relations between energy and quasimomenta. The key role in the obtained asymptotic approximation is played by librations, i.e., unstable periodic trajectories of the Hamiltonian system with an inverted potential. We also present an effective numerical algorithm for computing the widths of bands and discuss applications to quantum dimers.     

Adiabatic Evolution Generated by a Schrödinger Operator with Discrete and Continuous Spectra

In the paper, we consider the one-dimensional nonstationary Schrödinger equation with a potential slowly depending on time. It is assumed that the corresponding stationary operator depending on time as a parameter has a finite number of negative eigenvalues and absolutely continuous spectrum filling the positive semiaxis. A solution close at some moment to an eigenfunction of the stationary operator is studied. We describe its asymptotic behavior in the case where the eigenvalues of the stationary operator move to the edge of the continuous spectrum and, having reached it, disappear one after another.     

Asymptotics of the scattering phase for the Dirac operator: High energy, semi-classical and non-relativistic limits

In this paper we prove several results for the scattering phase (spectral shift function) related with perturbations of the electromagnetic field for the Dirac operator in the Euclidean space. Many accurate results are now available for perturbations of the Schrödinger operator, in the high energy regime or in the semi-classical regime. Here we extend these results to the Dirac operator. There are several technical problems to overcome because the Dirac operator is a system, its symbol is a 4×4 matrix, and its continuous spectrum has positive and negative values. We show that we can separate positive and negative energies to prove high energy asymptotic expansion and we construct a semi-classical Foldy-Wouthuysen transformation in the semi-classical case. We also prove an asymptotic expansion for the scattering phase when the speed of light tends to infinity (non-relativistic limit).     

Librations and ground-state splitting in a multidimensional double-well problem

We derive an asymptotic formula for the splitting of the lowest eigenvalues of the multidimensional Schrödinger operator with a symmetric double-well potential. Unlike the well-known formula of Maslov, Dobrokhotov, and Kolosoltsov, the obtained formula has the form A(h)e^?S/h(1 + o(1)), where S is the action on a periodic trajectory (libration) of the classical system with the inverted potential and not the action on the doubly asymptotic trajectory. In this expression, the principal term of the pre-exponential factor takes a more elegant form. In the derivation, we merely transform the asymptotic formulas in the mentioned work without going beyond the framework of classical mechanics.     

An application of the modified decomposition method for the solution of the coupled Klein–Gordon–Schrödinger equation

The modified decomposition method has been implemented for solving a coupled Klein–Gordon–Schrödinger equation. We consider a system of coupled Klein–Gordon–Schrödinger equation with appropriate initial values using the modified decomposition method. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. The numerical solutions of coupled Klein–Gordon–Schrödinger equation have been represented graphically.     

Nonlocal Darboux transformation of the two-dimensional stationary Schrödinger equation and its relation to the Moutard transformation

We consider a nonlocal Darboux transformation of the two-dimensional stationary Schrödinger equation and establish its relation to the Moutard transformation. We show that the Moutard transformation is a special case of the nonlocal Darboux transformation and obtain new examples of solvable two-dimensional stationary Schrödinger operators with smooth potentials as an application of the nonlocal Darboux transformation.     

On the spectrum of the Schrödinger operator perturbed by a rapidly oscillating potential

We study the spectrum of the one-dimensional Schrödinger operator perturbed by a rapidly oscillating potential. The oscillation period is a small parameter. We find explicitly the essential spectrum and study the existence of the discrete spectrum. Complete asymptotic expansions of the eigenvalues and corresponding eigenfunctions are constructed.