Gelfand-Levitan Theory Research Articles

Simultaneous uniqueness for the diffusion coefficient and initial value identification in a time-fractional diffusion equation

This article investigates the uniqueness of simultaneously determining the diffusion coefficient and initial value in a time-fractional diffusion equation with derivative order α∈(0,1). By additional boundary measurements and a priori assumption on the diffusion coefficient, the uniqueness of the eigenvalues and an associated integral equation for the diffusion coefficient are firstly established. The proof is based on the Laplace transform and the expansion of eigenfunctions for the solution to the initial value/boundary value problem. Furthermore, by using these two results, the simultaneous uniqueness in determining the diffusion coefficient and initial value is demonstrated from the Liouville transform and Gelfand–Levitan theory. The result shows that the uniqueness in simultaneous identification can be achieved, provided the initial values non-orthogonality to the eigenfunction of differential operators, which incorporates only one diffusion coefficient rather than scenarios involving two diffusion coefficients.

Reconstruction of the coefficients of a star graph from observations of its vertices

Consider a three-edge star graph, made up of unknown Sturm-Liouville operators on each edge. By using the heat propagation through the graph and measuring the heat transfer occurring at its vertices, we show that we can extract enough spectral data to reconstruct the three Sturm-Liouville operators by using the Gelfand-Levitan theory. Furthermore this reconstruction is achieved by a single measurement provided we use a special initial condition.

Construction of a Sturm-Liouville vessel using Gelfand-Levitan theory. On solution of the Korteweg-de Vries equation in the first quadrant

To study Korteweg-de Vries (KdV) equation, whose solution is based on the evolution of the Sturm-Liouville (SL) operator, we have recently developed scattering theory of the SL operator for an arbitrary analytic potential on the line. Here, we create a basis for solutions of the KdV of a different class, considering continuously differentiable potentials using the Gelfand-Levitan theory on the half line. The solution of the spectral problem for this operator is achieved by constructing a special object, invented by Livšic in the late 1970s, and called vessel. The central object of a vessel is a compact perturbation of a self-adjoint operator for which we develop a canonical form. Evolving the constructed vessel, we solve the KdV equation on the half line, coinciding with the given potential for t = 0. It is also shown that the initial value on the t-axis, or equivalently for x = 0, is prescribed by the choice of the spectral parameters, but can be perturbed using an “orthogonal” to the problem measure. The results, presented in this work, are following: (1) We address the KdV equation with initial values, satisfying Gelfand-Levitan theory assumptions, providing a detailed formula of the initial conditions for the x = 0, (2) we show that Nonlinear Shrodinger, canonical systems, and many more equations can be solved using theory of vessels, analogously to the Zacharov-Shabbath scheme, (3) we present a generalized inverse scattering theory on a line for potentials with singularities using prevessels, and (4) we present the tau function and its role in the solution of the problem.

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.

A trace formula and Schmincke inequality on the half-line

In this paper we derive a trace formula for the Schrodinger operator on the half-line. As a consequence we obtain a Schmincke type inequality with sharp constant. The main tool in our investigation is the inverse spectral Gelfand-Levitan theory, which allows us to compare two Schrodinger operators whose spectra differ by few eigenvalues.

The Gelfand-Levitan Theory Revisited

Gelfand and Levitan in their celebrated article in 1951, and later Gasymov and Levitan in 1964 have shown that a monotone increasing function is a spectral function of a singular Sturm-Liouville problem on a half-line in the limit point case at infinity if and only if it satisfies an existence and a smoothness condition. In this article, a closer look at the original statement reveals that the existence condition in fact follows from the smoothness condition which simplifies significantly the statement of the Gelfand-Levitan theory. We also provide two sufficient and verifiable conditions for a nondecreasing function to be the spectral function of a singular Sturm Liouville operator.

The approximation of the transmutation kernel

The transformation operator plays an important role in the direct and inverse spectral theory of Sturm-Liouville operators. In this paper we would like to approximate the kernel of the transformation operator used in the Gelfand-Levitan theory. The analytic properties of the solution allows for its representation by either a Taylor series about the diagonal or a Fourier cosine series. Example illustrating how the coefficients can be computed are provided.

Inverse Spectral Problem for the Laguerre Differential Operator

We construct a self-adjoint operator associated with the following given set of eigenvalues: {an+c}n≤−1∪{an+b}n≥0. This problem arises in sampling theory and deals with the construction of an interpolating basis associated with a given set of sampling points, the so-called two sided cardinal series. In order to solve this inverse spectral problem we shall need to extend the Gelfand–Levitan theory to operators close to the Laguerre differential operator.

Direct Computation of the Spectral Function

We would like to find an explicit formula for the spectral function of the following Sturm-Liouville problem: \[ \left \{ {\begin {array}{*{20}{c}} {Lf \equiv - \frac {{{d^2}}}{{d{x^2}}}f(x) + q(x)f(x),\quad x \geq 0,} \hfill \\ {f’(0) - mf(0) = 0.} \hfill \\ \end {array} } \right .\] A simple operational calculus argument will help us obtain an explicit formula for the transmutation kernel. The expression of the spectral function is then obtained through the nonlinear integral equation found in the Gelfand-Levitan theory.

Direct computation of the spectral function

We would like to find an explicit formula for the spectral function of the following Sturm-Liouville problem: \[ { L f ≡ − d 2 d x 2 f ( x ) + q ( x ) f ( x ) , x ≥ 0 , f ′ ( 0 ) − m f ( 0 ) = 0. \left \{ {\begin {array}{*{20}{c}} {Lf \equiv - \frac {{{d^2}}}{{d{x^2}}}f(x) + q(x)f(x),\quad x \geq 0,} \hfill \\ {f’(0) - mf(0) = 0.} \hfill \\ \end {array} } \right . \] A simple operational calculus argument will help us obtain an explicit formula for the transmutation kernel. The expression of the spectral function is then obtained through the nonlinear integral equation found in the Gelfand-Levitan theory.

Inversion theory for the magnetotelluric problem

Invertibility is established for the mathematical formulation most often used to describe the one-dimensional magnetotelluric problem. For a large space of conductivity functions (with no analyticity requirements), recovery of the conductivity function from magnetotelluric data is shown to be unique. This provides a basis to extend Weidelt's Gelfand-Levitan theory to a significantly more general setting.

Exact solution of the inverse acoustic scattering problem for the one-dimensional density profile in cylindrical geometry

The inhomogeneous acoustic wave equation in cylindrically symmetric geometry, is reduced to a one-dimensional Schrödinger equation for time harmonic waves. The scattering potential so formed is uniquely reconstructed by the Gelfand-Levitan theory. In physical terms, the scattering potential is a function of the impedance profile, which is then recovered. If the velocity profile is known (or constant) the density profile, as a function of the radial coordinate, can be deduced. A practical experiment is envisaged whereby the back-scattered time sequence is measured in a pulse-echo mode, and the data are used to reconstruct the density profile.

Impedance profile recovery from transmission data

We consider the problem of reconstructing the impedance profile from transmitted data. The problem (which arises for example in one‐dimensional tomography) is quite different from the problem in reflection seismology where the excitation and measurements are made at the surface. In our previous investigations, we used transmutation techniques to relate the transmitted data for time 0<t<∞ to the reflection data. This allows the use of the Gelfand–Levitan theory to recover the profile. In the present work, we consider the more difficult situation in which the transmitted data are given only up to a finite time. We study various maps and show that we can pose a fixed point problem to solve for the unknown profile. A new way of characterizing the Gelfand–Levitan theory for this problem based on minimization procedures is also presented.

Application of the Schur algorithm to the inverse problem for a layered acoustic medium

The Schur algorithm is a signal processing algorithm which works on a layer-stripping principle. Its time-domain version, the fast Cholesky recursion, is a fast and efficient algorithm well-suited for high-speed data processing. In this paper, these algorithms are applied to the inverse problem for a continuous layered acoustic medium. Three different excitations of the medium are considered: impulsive plane waves at normal incidence, impulsive plane waves at oblique incidence, and spherical waves emanating from an impulsive point source. The fast algorithms obtained for each of these problems seem to be computationally superior to past work done on these problems that employed Gelfand–Levitan theory to reconstruct the potential of a Schrödinger equation.

On the elastic profiles of a layered medium from reflection data. Part I. Plane-wave sources

It is shown that the density and the two Lamé profiles of a layered elastic medium can be uniquely determined from the impulse responses due to obliquely incident SH plane waves at two precritical angles of incidence and a normally incident P plane wave. A direct (noniterative) inversion algorithm is developed which construct the three elastic profiles of the layered medium from the reflection data. The equation of motion for an obliquely incident SH plane wave is transformed to the Schrödinger equation whose potential is related to the density profile, shear modulus profile, and the angle of incidence. The Gelfand–Levitan theory is used to recover the potential of the Schrödinger equation from the impulse response. When the impulse response is available at two angles of incidence, the shear modulus and density profiles can be separately obtained as a function of depth. It is further shown that the incompressibility profile can be obtained from the impulse response due to a normally incident P plane wave and the density and shear modulus profiles.

The Gelfand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems

In this paper, we are concerned with interpreting some classical inverse-scattering theories so that they are relevant to the one-dimensional inverse impulse-response problem which arises in reflection seismology First the Gelfand-Levitan integral equations (which arise in the inverse scattering theory for the Schrodinger equation) are derived strictly in the time domain. Originally these celebrated equations were derived as a means of solving an inverse spectral problem, which is naturally posed in the frequency domain. We show not only that these equations have a time-dependent formulation, but that their derivation is actually simpler here than in its original context. Next we give a similar derivation for the Marchenko integral equation. We then obtain, by an independent method, the integral equation of Gopinath and Sondhi, which is not unlike the Gelfand-Levitan linear equation. It was used by them as a means of solving a time-dependent inverse problem arising in speech synthesis. A new integral equation, similar to the Marchenko equation is also derived. Finally the integral equations are related to each other by direct transformation independent of the corresponding differential equations. The present paper opens with a section in which the equation of one-dimensional elastic waves and the corresponding seismic inverse impulse-response problem are transformed into a form to which the Gelfand-Levitan theory applies, and then into the equations which arose in Gopinath and Sondhi's work. We regard the integral equation of Gopinath and Sondhi as being directly applicable to the interpretation of seismic reflection data. The Gelfand-Levitan theory is also applicable but only after considerable transformation. Our calculations are entirely in the time domain. The resulting equations have the merit that in order to recover the unknown coefficient on a finite interval (0, L) we need to use the boundary data also only on a finite segment of the reflection time series. The length of the record is just the two-way travel time associated with length L. By contrast, frequency-domain approaches require that long records be Fourier transformed. We distinguish results which are true only when the unknown coefficient is continuous from those true when it is bounded but only piecewise continuous. In this work we have not addressed the important problem of deconvolution.

On the bound states in the continuum

It is shown that a special case of the amplitude modulation method of von Neumann and Wigner for constructing local potentials and normalizable wavefunctions of bound states in the continuum, and the formal use of the Gelfand-Levitan theory lead to the same results. A possible application for complex-energy resonance models is indicated.

Profile inversion of simple plasmas and nonuniform regions: Three-pole reflection coefficient

A mathematical method for the reconstruction of the electron density profile <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N(x)</tex> for an inhomogeneous, stratified, simple plasma is presented. If the reflection coefficient <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(k)</tex> of the incident probing electromagnetic wave is approximated by a third-order rational approximation, then profiles can be obtained which are similar to profiles obtained from simulated VHF satellite tracking data. The method is based upon the solution of the fundamental integral equation of inverse scattering (Gelfand-Levitan) theory. Using this theory it is possible to obtain an analytical expression for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N(x)</tex> as a function of distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> in the plasma if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(k)</tex> is a rational function of the wave number <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> . The integral equation is solved by the Laplace transform technique and checked by the differential operator technique. The method is exact once the functional form of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(k)</tex> is determined. Thus this analysis can supplement information about profiles which are obtained from calculations based on the WKB approximation (which approximation can also be applied to calculate the local wave impedance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W(x)</tex> for propagation in nonuniform regions). The functional characteristics of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N(x)</tex> depend on the pole positions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(k)</tex> in the complex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> plane. By calculating the variations in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N(x)</tex> due to variations in these pole positions, it is possible to set a finite error bound on the profile of the electron density if the error bound in the rational approximation to the reflection coefficient is known.

Bound-state models from Gelfand–Levitan theory

The Gelfand–Levitan formulation of the inverse problem is briefly reviewed and then used as a source of exactly solvable bound-state models which are suitable for examples in quantum mechanics courses. The case of one bound state is discussed in detail with emphasis on the role of the arbitrary normalization constant and its connection with physical parameters. An application to the deuteron is given. Possible pedagogical uses of these models are cited.

A method of the calculation of the mantle-electrical conductivity by the inversion of the geomagnetic secular variation data

An optimization technique is described for inverting the geomagnetic secular variation data to find out the electrical conductivity. It is simpler and more straightforward than the exact technique based on the Gelfand-Levitan theory of the inverse boundary value problem for the second-order differential equation [1].