Inverse Sturm-Liouville Problem Research Articles

The inverse Sturm-Liouville problem with generalized periodic boundary conditions

The inverse Sturm-Liouville problem with generalized periodic boundary conditions

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L 2[0, π], the Sturm-Liouville operator L D(y) = −y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L 2[0, π], i.e., q ∈ W 2 −1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L D is solved in the subspace of odd real functions σ(π/2 − x) = −σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.

On the properties of maps connected with inverse Sturm-Liouville problems

Let LD be the Sturm-Liouville operator generated by the differential expression Ly = −y″ + q(x)y on the finite interval [0, π] and by the Dirichlet boundary conditions. We assume that the potential q belongs to the Sobolev space W2ϑ[0, π] with some ϑ ≥ −1. It is well known that one can uniquely recover the potential q from the spectrum and the norming constants of the operator LD. In this paper, we construct special spaces of sequences ɫ2θ in which the regularized spectral data {sk}−∞∞ of the operator LD are placed. We prove the following main theorem: the map Fq = {sk} from W2ϑ to ɫ2θ is weakly nonlinear (i.e., it is a compact perturbation of a linear map). A similar result is obtained for the operator LDN generated by the same differential expression and the Dirichlet-Neumann boundary conditions. These results serve as a basis for solving the problem of uniform stability of recovering a potential. Note that this problem has not been considered in the literature. The uniform stability results are formulated here, but their proof will be presented elsewhere.

Solving an Inverse Sturm-Liouville Problem by a Lie-Group Method

Solving an inverse Sturm-Liouville problem requires a mathematical process to determine unknown function in the Sturm-Liouville operator from given data in addition to the boundary values. In this paper, we identify a Sturm-Liouville potential function by using the data of one eigenfunction and its corresponding eigenvalue, and identify a spatial-dependent unknown function of a Sturm-Liouville differential operator. The method we employ is to transform the inverse Sturm-Liouville problem into a parameter identification problem of a heat conduction equation. Then a Lie-group estimation method is developed to estimate the coefficients in a system of ordinary differential equations discretized from the heat conduction equation. Numerical tests confirm the accuracy and efficiency of present approach. Definite and random disturbances are also considered when comparing the present method with that by using a technique of numerical differentiation.

Numerical solution of forward and inverse Sturm–Liouville problems with an angular momentum singularity

This paper considers analytical and numerical-analytical issues encountered in the solution of an inverse Sturm–Liouville problem with a Bessel-type singularity. The main results are as follows: (i) there is a wide class of numerical methods for eigenvalue calculation which are not adversely affected by this type of singularity; (ii) a simple least-squares technique proposed by Röhrl in the regular case can actually be implemented in an even simpler way, and still work just as effectively; (iii) in some cases, a ‘unique local minimizer’ result of the type obtained by Röhrl is still available; (iv) finite spectral data and noise remain the most serious obstacles to accurate reconstruction.

Reconstruction method for inverse Sturm–Liouville problems with discontinuous potentials

In this paper, we present a method for solving inverse Sturm–Liouville problems by generalizing a Rundell–Sacks algorithm. The method is extended to deal with a general reference potential which can be adapted, e.g., to estimations of the jump-discontinuity points of the unknown potential. Moreover, its convergence properties are investigated. Numerical examples show that this modification can achieve more precise results from a given data set than the earlier method in using only the null reference potential and, therefore, the L2- and L∞-error can be reduced significantly.

Inverse problem for a singular differential operator

In this paper, we give the solution of the inverse Sturm–Liouville problem on two partially coinciding spectra. In particular, in this case we obtain Hochstadt’s theorem concerning the structure of the difference q ( x ) − q ̃ ( x ) for the singular Sturm Liouville problem defined on the finite interval ( 0 , π ) having the singularity type 1 4 sin 2 x at the points 0 and π .

Solvability of the inverse Sturm-Liouville problem with indecomposable boundary conditions

Solvability of the inverse Sturm-Liouville problem with indecomposable boundary conditions

A special case of the Sturm–Liouville inverse problem by three spectra: uniqueness results

A three spectral inverse Sturm–Liouville problem is considered arising in the theory of vibrating strings. It is shown that in some special cases the solution of this problem is unique.

A least-squares functional for solving inverse Sturm–Liouville problems

We present a variational algorithm for solving the classical inverse Sturm–Liouville problem in one dimension when two spectra are given. All critical points of the least-squares functional are at global minima, which suggest minimization by a (conjugate) gradient descent algorithm. Numerical examples show that the resulting algorithm works quite reliably without tuning for particular cases, even in the presence of noise. With the right choice of parameters, the minimization procedure converges very quickly.

Weak stability for an inverse Sturm–Liouville problem with finite spectral data and complex potential**This material is based upon work supported by the National Science Foundation under grant no DMS-0304280.

It is well known that knowing the Dirichlet–Dirichlet eigenvalues and the Dirichlet–Neumann eigenvalues determines uniquely the potential of a one-dimensional Schrödinger equation on a finite interval. We investigate here how well a potential may be approximated if only N of each type of eigenvalues are known to within an error ε.

On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

AbstractWe consider finite difference approximations of solutions of inverse Sturm‐Liouville problems in bounded intervals. Using three‐point finite difference schemes, we discretize the equations on so‐called optimal grids constructed as follows: For a staggered grid with 2 k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the Sturm‐Liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known.A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal one, which is unknown. Fortunately, as we show here, the grid dependence on the unknown coefficients is weak, so the inversion can be done on a precomputed grid for an a priori guess of the unknown coefficients. This observation leads to a simple yet efficient inversion algorithm, which gives coefficients that converge pointwise to the true solution as the number k of data points tends to infinity. The cornerstone of our convergence proof is showing that optimal grids provide an implicit, natural regularization of the inverse problem, by giving reconstructions with uniformly bounded total variation. The analysis is based on a novel, explicit perturbation analysis of Lanczos recursions and on a discrete Gel'fand‐Levitan formulation. © 2005 Wiley Periodicals, Inc.

Numerov's method for inverse Sturm–Liouville problems

This paper examines and extends a method, recently proposed by the author, for recovering from eigenvalues a symmetric potential of a Sturm–Liouville operator with Dirichlet boundary conditions. It uses Numerov's method and an extension by Andrew and Paine of an asymptotic correction technique of Paine, de Hoog and Anderssen. The method is extended to deal with natural boundary conditions and its convergence properties are investigated. Numerical results show that the method can extract more information from a given set of data than a related earlier method which uses a second-order discretization of the differential equation. Non-symmetric problems are also considered.

Equivalence of inverse Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter

Three inverse problems for a Sturm–Liouville boundary value problem − y″+ qy= λy, y(0)cos α= y′(0)sin α and y′(1)= f( λ) y(1) are considered for rational f. It is shown that the Weyl m-function uniquely determines α, f, and q, and is in turn uniquely determined by either two spectra from different values of α or by the Prüfer angle. For this it is necessary to produce direct results, of independent interest, on asymptotics and oscillation.

Eigenparameter Dependent Inverse Sturm-Liouville Problems

Uniqueness of and numerical techniques for the inverse Sturm-Liouville problem with eigenparameter dependent boundary conditions will be discussed. We will use a Gel'fand-Levitan technique to show that the potential q in can be uniquely determined using spectral data. In the presence of finite spectral data, q can be reconstructed using a successive approximation method that involves solving a hyperbolic boundary value problem that arises in the the Gel'fand-Levitan analysis. We also consider a shooting method where the right endpoint boundary condition is used in conjunction with a quasi-Newton scheme to recover the unknown potential, q.

Matrix Finite-Zone Dirac-Type Equations

Weyl–Titchmarsh matrix functions play an essential role in the spectral theory of Dirac-type equations ( Oper. Theory: Adv. Appl. 107 (1999)). In this paper, we have constructed a class of Weyl–Titchmarsh matrix-functions generating potentials of finite-zone type. It has turned out that the corresponding potentials have derivatives of an arbitrary order. Using the above-mentioned results, we deduce the matrix analogue of the trace formula for finite-zone matrix potentials. In the last part of the paper, we consider separately the scalar case of Dirac-type equations. For this case, we have constructed finite-zone potentials in explicit forms and proved that these potentials are quasiperiodical. We note that for scalar Schrödinger equations the corresponding results are well known (see Invent. Math. 30 (1975), 217–274; “Soliton and the Inverse Scattering Transform,” SIAM, Philadelphia, 1981; Rev. Sci. Technol. 23 (1983), 20–50; “Theory of Solitons, The Method of Inverse Problem,” New York, 1984; “Inverse Sturm–Liouville Problems,” VSP, Zeist, 1987; “Inverse Spectral Theory,” Academic Press, New York, 1987).

Inverse spectral problems for left-definite Sturm–Liouville equations with indefinite weight

Inverse Sturm–Liouville problems with indefinite weight are considered. Theorems analogous to those of both Hochstadt and Gelfand–Levitan are proved. As a sample result, under suitable symmetry conditions, we show that half of one spectrum suffices to determine the potential function uniquely.

Optimal finite difference grids for direct and inverse Sturm–Liouville problems

We study finite difference approximations of solutions of direct and inverseSturm–Liouville problems, in a finite or infinite interval on the real line. Thediscretization is done on optimal grids, with a three-point finite difference stencil.The optimal location of the grid points is calculated via a rational approximationof the Neumann-to-Dirichlet map and the latter converges exponentiallyfast. We prove that optimal grids obtained for constant coefficients areasymptotically optimal for variable coefficient direct problems. We also showthat optimal grids, together with methods of inverse spectral problemsfor Jacobi matrices, can be used for the solution of continuous inverseSturm–Liouville problems. In particular, we formulate and analyse a new inversionalgorithm, where the unknown coefficients that we image are optimallydiscretized. We prove that optimal grids provide necessary conditionsfor convergence of the discrete inverse problem and we demonstrate theeffectiveness of our imaging approach through numerical simulations.

Inverse Spectral Problems for Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions

Inverse Sturm–Liouville problems with eigenparameter-dependent boundary conditions are considered. Theorems analogous to those of both Hochstadt and Gelfand and Levitan are proved. In particular, let ly = (1/r)(−(py′)′+qy), l ˜ y = ( 1 / r ˜ ) ( - ( p ˜ y ′ ) ′ + q ˜ y ) , Δ = [ a b c d ] and ∑ = [ r s t u ] where det Δ = δ > 0, c ≠ 0, det ∑ > 0, t ≠ 0 and (cs + dr − au − tb)2 < 4(cr − ta)(ds − ub). Denote by (l; α; Δ) the eigenvalue problem ly = λy with boundary conditions y(0)cosα+y′(0)sinα = 0 and (aλ+b)y(1) = (cλ+d)(py′)(1). Define ( l ˜ ; α; Δ) as above but with l replaced by l ˜ . Let wn denote the eigenfunction of (l; α; Δ) having eigenvalue λn and initial conditions wn(0) = sin α and pw′n(0) = −cos α and let γn = −awn(1)+cpw′n(1). Define w ˜ n and γ ˜ n similarly. As sample results, it is proved that if (l; α; Δ) and ( l ˜ ; α; Δ) have the same spectrum, and (l; α; Σ) and ( l ˜ ; α; Σ) have the same spectrum or ∫ | w n | 0 1 r d t + ( | γ n | 2 / δ ) = ∫ | w ˜ n | 0 1 r ˜ d t + ( | γ ˜ n | 2 / δ ) for all n, then q/r = q ˜ / r ˜ .

An inverse sturm-Liouville problem by three spectra

It is shown that the potential of the Sturm-Liouville equation on interval [0,a] may be restored by the spectra of three boundary problems generated by the equation on the intervals [0,a], [0, 1/2a] and [1/2a,a], respectively. The algorithm of construction is given as well as the sufficient conditions for three sequences of real numbers to be the spectra of the mentioned boundary problems. The problem on [0,a] describes small vibrations of a smooth string with fixed ends. The problems on the half-intervals describe vibrations of the same string clamped at the point of equilibrium.