Non-self-adjoint sturm-liouville operators with matrix potentials

We obtain asymptotic formulas for non-self-adjoint operators generated by the Sturm-Liouville system and quasiperiodic boundary conditions. Using these asymptotic formulas, we obtain conditions on the potential for which the system of root vectors of the operator under consideration forms a Riesz basis.     

On the Riesz basis property of the root functions of a discontinuous boundary problem

In this paper, we consider the nonself‐adjoint discontinuous Sturm Liouville operator with periodic (antiperiodic) boundary condition and compatibility conditions. Asymptotic formulas of eigenvalues and eigenfunctions of the operator are obtained. Using these accurate asymptotic formulas for eigenvalues and eigenfunctions, we prove the basisness of the root functions of the boundary value problem.     

A One-Dimensional Schrödinger Operator with Square-Integrable Potential

We study the spectral properties of a one-dimensional Schrödinger operator with squareintegrable potential whose domain is defined by the Dirichlet boundary conditions. The main results are concerned with the asymptotics of the eigenvalues, the asymptotic behavior of the operator semigroup generated by the negative of the differential operator under consideration. Moreover, we derive deviation estimates for the spectral projections and estimates for the equiconvergence of the spectral decompositions. Our asymptotic formulas for eigenvalues refine the well-known ones.     

On non-self-adjoint Sturm-Liouville operators in the space of vector functions

In this article we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operator generated in L ₂ ^m [0, 1] by the Sturm-Liouville equation with m × m matrix potential and the boundary conditions which, in the scalar case (m = 1), are strongly regular. Using these asymptotic formulas, we find a condition on the potential for which the root functions of this operator form a Riesz basis.     

On the basis property of the root functions of differential operators with matrix coefficients

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.     

On the nonself-adjoint ordinary differential operators with periodic boundary conditions

In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the nonself-adjoint ordinary differential operator with periodic and antiperiodic boundary conditions, when coefficients are arbitrary summable complex-valued functions. Then using these asymptotic formulas, we obtain necessary and sufficient conditions on the coefficient for which the root functions of these operators form a Riesz basis.     

Semiclassical asymptotic spectrum of a Hartree-type operator near the upper boundary of spectral clusters

We consider the problem for eigenvalues of a perturbed two-dimensional oscillator in the case of a resonance frequency. The exciting potential is given by a Hartree-type integral operator with a smooth self-action potential. We find asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundary of spectral clusters, which form around energy levels of the nonperturbed operator. To calculate them, we use asymptotic formulas for quantum means.     

Asymptotic analysis of non-self-adjoint Hill operators

We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L _t (q) with a potential q ∈ L ₁[0,1] and t-periodic boundary conditions, t ∈ (?π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L ₂(?∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies sufficient conditions.     

On the riesz basisness of the root functions of the nonself-adjoint sturm-liouville operator

In this article we obtain the asymptotic formulas for eigenfunctions and eigenvalues of the nonself-adjoint Sturm-Liouville operators with periodic and antiperiodic boundary conditions, when the potential is an arbitrary summable complex-valued function. Then using these asymptotic formulas, we find the conditions on Fourier coefficients of the potential for which the eigenfunctions and associated functions of these operators form a Riesz basis inL ₂(0, 1).     

Uniform asymptotics of the eigenvalues and eigenfunctions of the Dirac system with an integrable potential

We consider the Dirac operator on a finite interval with a potential belonging to some set X completely bounded in the space L₁[0, π] and with strongly regular boundary conditions. We derive asymptotic formulas for the eigenvalues and eigenfunctions of the operator; moreover, the constants occurring in the estimates for the remainders depend on the boundary conditions and the set X alone.     

Asymptotic regularity of solutions to Hartree–Fock equations with Coulomb potential

We study the asymptotic regularity of solutions to Hartree–Fock (HF) equations for Coulomb systems. To deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo‐differential operators on conical manifolds. First, the non‐self‐consistent‐field case is considered, which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator, which guarantee ellipticity for the local part of the Fock operator on the open stretched cone ?₊ × S². This proves the existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed‐point approach based on Cancès and Le Bris analysis of the level‐shifting algorithm, we show via another bootstrap argument that the corresponding self‐consistent‐field solutions to the HF equation have the same type of asymptotic regularity. Copyright © 2008 John Wiley & Sons, Ltd.     

Asymptotic formulas for fundamental solutions of the Dirac system with complex-valued integrable potential

We study the Dirac operator with a complex-valued integrable potential in the space ? = L ₂[0, π]⊕L ₂[0, π]. We obtain asymptotic formulas for a fundamental solution system of an operator. Remainders in each of the formulas are estimated.     

On the Asymptotics of Eigenvalues of a Fourth-Order Differential Operator with Matrix Coefficients

We study a fourth-order differential operator with matrix coefficients whose domain is determined by the Dirichlet boundary conditions. An asymptotics of the weighted average of the eigenvalues of this operator is obtained in the general case. As a consequence of this result, we present the asymptotics of the eigenvalues in several special cases. The obtained results significantly improve the already known asymptotic formulas for the eigenvalues of a one-dimensional fourth-order differential operator.     

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.     

Elliptic and parabolic problems in unbounded domains

We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semi‐explicit representation formulas via operator‐valued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On sharp asymptotic formulas for the Sturm–Liouville operator with a matrix potential

In this article we obtain the sharp asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operators generated by a system of the Sturm–Liouville equations with Dirichlet and Neumann boundary conditions. Using these asymptotic formulas, we find a condition on the potential for which the root functions of these operators form a Riesz basis.     

Asymptotics of the spectrum of the Sturm-Liouville operator with regular boundary conditions

On the interval (0, π), we consider the spectral problem generated by the Sturm-Liouville operator with regular but not strongly regular boundary conditions. For an arbitrary potential q(x) ∈ L ₁ (0, π) [q(x) ∈ L ₂(0, π)], we establish exact asymptotic formulas for the eigenvalues of this problem.     

A class of second‐order differential operators with eigenparameter‐dependent boundary and transmission conditions

In this study, we investigate a Sturm–Liouville type problem with eigenparameter‐dependent boundary conditions and eigenparameter‐dependent transmission conditions. By establishing a new self‐adjoint operator A associated with the problem, we construct fundamental solutions and obtain asymptotic formulae for its eigenvalues and fundamental solutions. Also we investigate some properties of its spectrum. Copyright © 2013 John Wiley & Sons, Ltd.     

On the asymptotics of eigenfunctions of Sturm-Liouville operators with distribution potentials

We consider a Sturm-Liouville operator in the space L ₂[0, π] and derive asymptotic formulas for the eigenvalues and eigenfunctions of this operator for the case of Dirichlet-Neumann boundary conditions. The leading and second terms of the asymptotics are found in closed form.     

Tunnel splitting of the spectrum and bilocalization of eigenfunctions in an asymmetric double well

We consider the one-dimensional stationary Schrödinger equation with a smooth double-well potential. We obtain a criterion for the double localization of wave functions, exponential splitting of energy levels, and the tunneling transport of a particle in an asymmetric potential and also obtain asymptotic formulas for the energy splitting that generalize the formulas known in the case of a mirror-symmetric potential. We consider the case of higher energy levels and the case of energies close to the potential minimums. We present an example of tunneling transport in an asymmetric double well and also consider the problem of tunnel perturbation of the discrete spectrum of the Schrödinger operator with a single-well potential. Exponentially small perturbations of the energies occur in the case of local potential deformations concentrated only in the classically forbidden region. We also calculate the leading term of the asymptotic expansion of the tunnel perturbation of the spectrum.