Regularized Trace Formula Research Articles

Trace Formula on Transmission Eigenvalues of the Sturm–Liouville Problem

In this paper, we consider eigenvalues and traces of a special Sturm-Liouville problem, which originates from an acoustic scattering problem with a spherically symmetric speed of sound. Regularized trace formulae on transmission eigenvalues of the Sturm-Liouville problem are obtained.

Spectral asymptotics and regularized traces for Dirac operators on a star-shaped graph

In this article, we consider the spectral problems for Dirac operators on a star-shaped graph. First, we derive the asymptotic expressions of the eigenvalues and then explicitly establish the regularized trace formulae for the operators in terms of the coefficients of the operators.

The second regularized trace of differential operator equation with the semi‐periodic boundary conditions

AbstractIn this paper, we have obtained the second regularized trace formula for the differential operator equation with the semi‐periodic boundary conditions. Copyright © 2012 John Wiley & Sons, Ltd.

The Regularized Trace Formula of the Spectrum of a Dirichlet Boundary Value Problem with Turning Point

We calculate the regularized trace formula of the infinite sequence of eigenvalues for some version of a Dirichlet boundary value problem with turning points.

Regularized Trace for Sturm–Liouville Differential Operator on a Star-Shaped Graph

The sum of the eigenvalues {λn} of an operator is usually called its trace. For the eigenvalues λn of an differential operator, the series \({\sum_n \lambda_n}\) , generally speaking, diverges; however, it can be regularized by subtracting from λn the first terms of the asymptotic expansion, which interfere with the convergence of the series. The sum of such a regularized series is called the trace. In this work, we consider the spectral problem for Sturm–Liouville differential operator on d-star-type graph with a Kirchhoff-type condition in the internal vertex, where the integer d ≥ 2. Regularized trace formula of this operator is established with residue techniques in complex analysis.

Study of the asymptotic eigenvalue distribution and trace formula of a second order operator-differential equation

The purpose of writing this article is to show some spectral properties of the Bessel operator equation, with spectral parameter-dependent boundary condition. This problem arises upon separation of variables in heat or wave equations, when one of the boundary conditions contains partial derivative with respect to time. To illustrate the problem and the proof in detail, as a first step, the corresponding operator’s discreteness of the spectrum is proved. Then, the nature of the eigenvalue distribution is established. Finally, based on these results, a regularized trace formula for the eigenvalues is obtained. MSC: 34B05; 34G20; 34L20; 34L05; 47A05; 47A10.

Erratum: “New trace formulae for a quadratic pencil of the Schrödinger operator” [J. Math. Phys. 51, 033506 (2010)

In this note we correct a mistake made in my paper [J. Math. Phys. 51, 033506 (2010)]. Using the contour integration method we shall obtain a new regularized trace formula for Schrödinger operator with energy-dependent potential.

The second regularized trace of a self adjoint differential operator given in a finite interval with bounded operator coefficient

In this work a formula for the second regularized trace of a self adjoint differential operator which is given in a finite interval and with bounded operator coefficient was found.

A regularized trace formula for second order differential operator equations

In this paper, we deal with abstract Sturm-Liouville problems when the potential of the differential equation is an operator function in a Hilbert space $H$. We generalize trace formula obtained by [7], [9] for the classic regular Sturm-Liouville problems. We investigate the spectrum and obtained a regularized trace formula for the Sturm-Liouville operator with an operator coefficient.

A Regularized Trace Formula for Differential Equations with Trace Class Operator Coefficients

A Regularized Trace Formula for Differential Equations with Trace Class Operator Coefficients

The third regularized trace formula for the Sturm–Liouville operator with the bounded operator coefficient

In this work, we obtain a regularized trace formula for the Sturm–Liouville operator with a bounded operator coefficient given in the finite interval.

New trace formulae for a quadratic pencil of the Schrödinger operator

This work deals with the eigenvalue problem for a quadratic pencil of the Schrödinger operator on a finite closed interval with the two-point boundary conditions. We will obtain new regularized trace formulas for this class of differential pencil.

Trace formulae for Schrödinger systems on graphs

For Schrödinger systems on metric graphs with \delta'-type conditions at the central vertex, firstly, we obtain precise description for the square root of the large eigenvalue up to the o(1/n)-term. Secondly, the regularized trace formulae for Schrödinger systems are calculated with some techniques in classical analysis. Finally, these formulae are used to obtain a result of inverse problem in the spirit of Ambarzumyan.

A trace formula of a boundary value problem for the operator Sturm-Liouville equation

We obtain a regularized trace formula for the operator Sturm-Liouville equation with a boundary condition depending on a spectral parameter.

The regularized trace of a linear ordinary differential operator with polynomial coefficients on l 2(−∞, +∞)

Regularized trace formulas for a class of linear ordinary differential operators on the real line with polynomial coefficients are obtained. The proof uses matrix representations of such operators and general methods of perturbation theory for obtaining regularized trace formulas for abstract operators on Hilbert spaces.

Higher order regularized trace formula for the regular Sturm–Liouville equation contained spectral parameter in the boundary condition

In this paper we have obtained a formula for order ⩾2 regularized trace for a regular Sturm–Liouville equation with operator coefficient and contains spectral parameter in the boundary condition.

Non-nuclear perturbations of discrete operators and trace formulae

A trace formula is obtained for unbounded discrete operators perturbed by a Hilbert-Schmidt operator; this formula may be called the discrete analogue of M. Krein's formula for nuclear perturbations. A regularized trace formula of Krein's type is also proved for perturbations in the class , , for arbitrary compact and relatively compact perturbations depending on the behaviour at infinity of the distribution function of the spectrum of the unperturbed operator.

The Spectrum of Resonances and the Trace Formula in a Potential Scattering Problem

For the wave equation with a potential perturbation, the localization and the asymptotic distribution of resonances, i.e., poles of the scattering matrix, are studied. To this end, it proves useful to exploit the relation between these poles and the spectrum of the corresponding Lax–Phillips semigroup. An explicit description of the generator of this semigroup is given. The regularized trace formula for the operators specifying the evolution of the initial data is applied to estimate how rarefied the spectrum of resonances can be.

Traces of operators with relatively compact perturbations

Regularized trace formulae are proved for abstract operators with a perturbing operator subordinate to the non-perturbed operator in the following sense: is a compact operator in some Schatten-von Neumann class of finite order. Two essentially different cases can be distinguished here: the resolvent of is either of trace class or not. Five theorems describing various cases of operator subordination and the structure of the spectrum of the unperturbed operator are proved.

An Approximate Regularized Trace Formula for an Ordinary Fourth-Order Differential Operator

An Approximate Regularized Trace Formula for an Ordinary Fourth-Order Differential Operator