Krein's Method Research Articles

On Justification of the Gelfand–Levitan–Krein Method for a Two-Dimensional Inverse Problem

On Justification of the Gelfand–Levitan–Krein Method for a Two-Dimensional Inverse Problem

On justification of the Gelfand–Levitan–Krein method for a two-dimensional inverse problem

On justification of the Gelfand–Levitan–Krein method for a two-dimensional inverse problem

Particle creation with excited de Sitter modes

Recently, we introduced exited de Sitter modes to study the power spectrum that was finite in Krein space quantization and the trans-Plankian corrections because of the exited mode being nonlinear (M. Mohsenzadeh et al. Eur. Phys. J. C, 74, 2920 (2014) doi:10.1140/epjc/s10052-014-2920-5 ). It was shown that the de Sitter limit of corrections reduces to that obtained via several previous conventional methods; moreover, with such modes the space–time symmetry becomes manifest. In this paper, inspired by the Krein method and using exited de Sitter modes as the fundamental initial states during the inflation, we calculate particle creation in the spatially flat Robertson–Walker space–time. It is shown that in de Sitter and Minkowski space–time in the far past time limit, our results coincide with the standard results.

Inverse spectral problem for a rod with multiple cracks

A problem of identification of multiple cracks in a rod using spectral data is considered. The cracks are simulated by translational springs. A method for determination of the number of springs, their locations and flexibilities is developed. The problem is reduced to the problem which can be solved by Krein׳s method. The Krein method enables to reconstruct the damage parameters by means of the use of natural frequencies for longitudinal vibration of the rod with free–free and fixed–free end conditions.

The Krein method and the globally convergent method for experimental data

Comparison of numerical performances of two methods for coefficient inverse problems is described. The first one is the classical Krein integral equation method, and the second one is the recently developed approximately globally convergent numerical method. This comparison is performed for both computationally simulated and experimental data.

Mark Krein's method of directing functionals and singular potentials

AbstractIt is shown that M. Krein's method of directing functionals can be used to prove the existence of a scalar spectral measure for certain Sturm–Liouville equations with two singular endpoints. The essential assumption is the existence of a solution of the equation that is square integrable at one singular endpoint and depends analytically on the eigenvalue parameter.

Schrödinger and Dirac operators with the Aharonov–Bohm and magnetic-solenoid fields

We construct all self-adjoint Schrödinger and Dirac operators (Hamiltonians) with both the pure Aharonov–Bohm (AB) field and the so-called magnetic-solenoid field (a collinear superposition of the AB field and a constant magnetic field). We perform a spectral analysis for these operators, which includes finding spectra and spectral decompositions, or inversion formulae. In constructing the Hamiltonians and performing their spectral analysis, we follow, respectively, the von Neumann theory of self-adjoint extensions of symmetric operators and the Krein method of guiding functionals.

Self-adjoint extensions and spectral analysis in the generalized Kratzer problem

We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional non-relativistic motion of a particle in the potential field , For g2>0 and g1<0, the potential is known as the Kratzer potentialVK(x) and is usually used to describe molecular energy and structure, interactions between different molecules and interactions between non-bonded atoms. We construct all self-adjoint Schrödinger operators with the potential V(x) and represent rigorous solutions of the corresponding spectral problems. Solving the first part of the problem, we use a method of specifying self-adjoint extensions by (asymptotic) self-adjoint boundary conditions. Solving spectral problems, we follow Krein's method of guiding functionals. This work is a continuation of our previous works devoted to the Coulomb, Calogero and Aharonov–Bohm potentials.

Numerical algorithm for two-dimensional inverse acoustic problem based on Gel'fand–Levitan–Krein equation

Abstract The inverse problem for the acoustic equation is considered. We propose a method of reconstruction of the density approximating 2D inverse acoustic problem by a finite system of one dimensional inverse acoustic problems. The 2D analogy of the Gel'fand–Levitan–Krein method is established. The inverse acoustic problem is formulated and the short outline of the history and development in this field are given in Section 1. In Section 2 we consider the 2D analogy of the Gel'fand–Levitan–Krein equation. The N-approximation of the Gel'fand–Levitan–Krein equation is obtained for inverse acoustic problem in Section 3. The numerical results are presented in Section 4.

The boundary control approach to inverse spectral theory

We establish connections between different approaches to inverse spectral problems: the classical Gelfand–Levitan theory, the Krein method, the Simon theory, the approach proposed by Remling and the boundary control method. We show that the boundary control approach provides simple and physically motivated proofs of the central results of other theories. We demonstrate also the connections between the dynamical and spectral data and derive the local version of the classical Gelfand–Levitan equations.

The Dirac Hamiltonian with a superstrong Coulomb field

We consider the quantum mechanical problem of a relativistic Dirac particle moving in the Coulomb field of a point charge Ze. It is often declared in the literature that a quantum mechanical description of such a system does not exist for charge values exceeding the so-called critical charge with Z = α−1 = 137 because the standard expression for the lower bound-state energy yields complex values at overcritical charges. We show that from the mathematical standpoint, there is no problem in defining a self-adjoint Hamiltonian for any charge value. Furthermore, the transition through the critical charge does not lead to any qualitative changes in the mathematical description of the system. A specific feature of overcritical charges is a nonuniqueness of the self-adjoint Hamiltonian, but this nonuniqueness is also characteristic for charge values less than critical (and larger than the subcritical charge with $$Z = (\sqrt 3 /2)\alpha ^{ - 1} = 118$$ ). We present the spectra and (generalized) eigenfunctions for all self-adjoint Hamiltonians. We use the methods of the theory of self-adjoint extensions of symmetric operators and the Krein method of guiding functionals. The relation of the constructed one-particle quantum mechanics to the real physics of electrons in superstrong Coulomb fields where multiparticle effects may be crucially important is an open question.

Boundary control and Gel'fand–Levitan–Krein methods in inverse acoustic problem

In this paper we study the inverse problem for the acoustic equation. The boundary control method (BC-method) and Gel'fand—Levitan—Krein method (GLK-method) are investigated and compared. In Section 1 inverse acoustic problem is formulated. In Section 2 we consider the scheme of BC-method. In Section 3 the multidimensional analog of Gel'fand—Levitan—Krein equation is obtained for inverse acoustic problem. In Section 4 we investigate discrete analogs of main relations of both methods. In Section 5 we consider the numerical results.

Boundary control and Gel'fand–Levitan–Krein methods in inverse acoustic problem

In this paper we study the inverse problem for the acoustic equation. The boundary control method (BC-method) and Gel'fand—Levitan—Krein method (GLK-method) are investigated and compared. In Section 1 inverse acoustic problem is formulated. In Section 2 we consider the scheme of BC-method. In Section 3 the multidimensional analog of Gel'fand—Levitan—Krein equation is obtained for inverse acoustic problem. In Section 4 we investigate discrete analogs of main relations of both methods. In Section 5 we consider the numerical results.

On integral representation of the continuous O(n)-positive definite functions

The aim of this paper is to study integral representation for continuous O(n)-positive definite functions. Using the Krein method of directing functionals we prove the theorem about integral representation, give the criteria of uniqueness of this representation and description of all representations in the case of ununiqueness. On the basis of these results we solve analogous problems for continuous radial positive definite functions.

Krein's method with certain singular kernel for solving the integral equation of the first kind

In this paper, we are going to obtain the spectral relations for the Fredholm integral equation of the first kind with certain singular kernel, by using Krein's method.

Extremal positive and self-adjoint extensions of suboperators

F. ollowing Halmos [1] a suboperator is a map from a subspace of a (complex) Hilbert space into the whole space which is a restriction of some bounded linear transformation, an operator on the space. A suboperator is said to be subpositive, subself-adjoint (and so on) if it is a restriction of a positive, self-adjoint (and so on) operator of the space. The characterization problem of subself-adjoint suboperators is solved by Krein, see [2]. Krein shows that a bounded and symmetric linear map from a subspace of a Hilbert space has a symmetric extension to the whole space (hence a self-adjoint extension) with the same bound. Krein's method of proof yields the normpreserving smallest, resp. largest self-adjoint extensions, in the usual ordering of self-adjoint operators on Hilbert space, too. The extension theorem for subpositive suboperators, due to one of the authors [3], led naturally to the reduction of Krein's theorem to the just mentioned positive extendibility problem. Another characterization of subpositive suboperators is given by Halmos [1]. The purpose of this note is first to show that the extension of a subpositive suboperator constructed in [3] is of minimal norm and smallest in the ordering of self-adjoint operators, between all the positive extensions. The existence of a largest positive extension (with the same norm) is then a simple consequence of the existence of the smallest positive extension. The extremal norm-preserving extension problem with respect to subself-adjoint suboperators is thus reduced to the corresponding, just mentioned question for subpositive suboperators.

Application of the Krein's method for determination of natural frequencies of periodically supported beam based on simplified Bresse-Timoshenko equations

Free vibration of a periodically supported (multispan) beamin via a simplified Bresse-Timoshenko theory is studied by the Krein's method suggested in 1933 for the Bernoulli-Euler beams. Approximate differential equations are utilized with both shear deformations and rotary inertia included, but with the term representing the joint action of these effect omitted. Detailed analytical and numerical analysis are performed for the natural frequencies of beams with different boundary conditions at their ends. Following Krein, the continuity requirements at the intermediate supports are treated as equations in finite differences and solved exactly.

A generalization of M. G. Krein's method of directing functional to linear relations

SynopsisM. G. Krein's method of directing functional is generalized in a straight-forward way to symmetric linear relations. Applications to Stieltjes differential boundary problems are indicated.

Reconstruction of the Schr�dinger-equation potential from scattering data by the Krein method

Reconstruction of the Schrodinger-equation potential (for the case of s-scattering) from scattering data by the Krein method is discussed. Analytically, the problem reduces to the solution of a system of linear inhomogeneous algebraic equations for certain functions. The Bargmann potentials, determined earlier by other methods, are shown to result from the solution of the problem for various particular cases.