Regularized Trace Formula Research Articles

Trace regularization problem for a fourth‐order differential operator on separable Banach space

In this study, we employ new results on the semi‐inner product, the adjoint operator, and the Schatten class of operators on a separable Banach space, which allow us to investigate a differential operator with unbounded operator‐valued coefficients. Specifically, we derive an asymptotic formula for the second regularized trace based on the asymptotic and spectral properties of the extended operator. For this purpose, we use the theory of continuous dense embeddings and known results regarding the regularized trace in Hilbert space. We conclude our paper by providing examples that support our findings.

Formula for regularized trace of 2m-order differential operator with periodic boundary conditions

We obtain a regularized trace formula for 2m-order differential operator perturbed by a quasi-differential perturbation and with periodic boundary conditions.

THE REGULARIZED TRACE FORMULA FOR A DISCONTINUOUS STURM-LIOUVILLE PROBLEM WITH A SPECTRAL PARAMETER DEPENDENT JUMP CONDITION

The first regularized trace formula for a discontinuous Sturm-Liouville problem with a spectral parameter dependent jump condition is obtained.

Second regularized trace of a differential operator of Sturm–Liouville type

We consider a Sturm–Liouville‐type differential operator defined on a separable Hilbert space with bounded operator coefficient and periodic boundary conditions given on a finite interval. We obtain an explicit formula which we call the second regularized trace formula of the mentioned operator.

REGULARIZED TRACE OF OPERATOR WITH UNBOUNDED COEFFICIENT

Abstract. This paper is devoted to the study of the regularized trace formula of an operator generated by a differential expression with operator coefficient and eigenvalue dependent boundary condition.

Trace formula of the differential operator with delays on a quantum graph

In this paper a class of differential operators with retarded arguments on a lasso graph is studied. We derive the asymptotic expressions of its large eigenvalues and obtain a new regularized trace formula for this class of differential operators.

On the Spectral Properties of a Fourth-Order Self-Adjoint Operator

We consider a spectral problem for a fourth-order differential operator with real periodic coefficients and Neumann-type boundary conditions. For this operator, the eigenvalue asymptotics and a regularized trace formula are obtained.

The regularized trace formula for ”weighted” Sturm-Liouville equation with δ-interaction

The regularized trace formula of first order for the ”weighted” Sturm-Liouville equation with point δ-interaction is obtained.

Trace formulae for a conformable fractional diffusion operator

In this paper, we obtain the regularized trace formulae for a diffusion operator, which includes conformable fractional derivatives of order ? (0 < ? ? 1) instead of the ordinary derivatives in a traditional diffusion operator by the contour integration method. The results of this paper are of great importance in solving inverse problems and can be considered as partial fractional generalizations.

A regularized trace formula for ”weighted” Sturm-Liouville equation with point δ- interaction

In this study, we obtain a formula for the regularized trace formula for "weighted" Sturm--Liouville equation with point $\delta $- interaction. At the begining, for the correct determination of solutions of analyzed equation at the point of discontinuty, the matching conditions are required. As a result, an equation is derived for the eigenvalues of the differential operator given in this study.

A Second Regularized Trace Formula for a Fourth Order Differential Operator

In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system; for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator.

On Higher Regularized Traces of a Differential Operator with Bounded Operator Coefficient Given in a Finite Interval

In this work, we find a higher regularized trace formula for a regular Sturm–Liouville differential operator with operator coefficient.

A second regularized trace formula for a higher order differential operator

In this paper, we obtain a second regularized trace formula onL2([0, π]; H)for a higher order self‐adjoint differential operator with unbounded operator‐valued coefficient, whereHis a separable Hilbert space.

Regularized Trace of a Sturm–Liouville Operator on a Curve with a Regular Singularity on the Chord

For a Sturm–Liouville operator on a piecewise smooth curve, we study the effect that the spectrum of a nonintegrable singularity of the potential on the segment joining the endpoints of this curve has on the operator asymptotics. It is shown that in the case where the singular point does not give rise to the branching of solutions in its neighborhood (the case of trivial monodromy), the spectral asymptotics and the formula for the regularized trace look the same as for the classical Sturm–Liouville operator on a segment with smooth potential. Further, it is shown that in the case of nontrivial monodromy the spectral asymptotics substantially depends on the commensurability of the parts into which the segment is partitioned by the singular point $$\theta $$ : if $$\theta $$ is rational, then the spectrum is divided into finitely many series, each going to infinity along “its own” parabola. In this case, the regularized trace formula is significantly more complicated and does not show any similarity with the classical formula.

New Trace Formulae for Sturm–Liouville Operators on the Lasso-Graph

The Sturm–Liouville operators on the lasso graph are considered. We obtain new regularized trace formulae for this class of differential operators, which are very useful to further study the inverse spectral problems of this kind of operators.

Traces for Sturm–Liouville Operators with Frozen Argument on Star Graphs

In this paper, Sturm-Liouville operators with frozen argument on star graphs are studied. First, we derive the asymptotic expression for the large eigenvalues. Second, the regularized trace formulae for these operators are calculated with the method of complex analysis.

Essentially isospectral transformations and their applications

We define and study the properties of Darboux-type transformations between Sturm--Liouville problems with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.

Traces and inverse nodal problem for Sturm–Liouville operators with frozen argument

In this paper, Sturm–Liouville operators with frozen argument are studied. We derive the regularized trace formulae for these operators. Also we provide asymptotic expressions for the nodal points and give a constructive procedure for solving an inverse nodal problem.

The formula for the regularized trace of the Sturm-Liouville operator with a logarithmic potential

We have obtained a regularized trace formula for the Sturm-Liouville operator on a semi-axis with a logarithmic potential.

Traces and inverse nodal problems for Dirac-type integro-differential operators on a graph

In this paper, Dirac-type integro-differential operators on the star graph are studied. First, we derive the asymptotic expressions for the large eigenvalues. Second, the regularized trace formulae are obtained for these operators. We also provide asymptotic expressions for the nodal points and give a constructive procedure for solving an inverse nodal problem.