We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as $X_1$-Jacobi and $X_1$-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the the compact interval $[-1,1]$ or the half-line $[0,\infty)$, respectively, and they are a basis of the corresponding $L^2$ Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second order operator has a complete set of polynomial eigenfunctions $\{p_i\}_{i=1}^\infty$, then it must be either the $X_1$-Jacobi or the $X_1$-Laguerre Sturm-Liouville problem. A Rodrigues-type formula can be derived for both of the $X_1$ polynomial sequences.
In this paper, we formulate a regular $q$-fractional Sturm--Liouville problem (qFSLP) which includes the left-sided Riemann--Liouville and the right-sided Caputo $q$-fractional derivatives of the same order $α$, $α\in (0,1)$. We introduce the essential $q$-fractional variational analysis needed in proving the existence of a countable set of real eigenvalues and associated orthogonal eigenfunctions for the regular qFSLP when $α>1/2$ associated with the boundary condition $y(0)=y(a)=0$. A criteria for the first eigenvalue is proved. Examples are included. These results are a generalization of th…
We describe the wave functional model for the minimal (symmetric) Sturm-Liouville operator on the finite interval. We construct the wave spectrum of this operator, then, following the abstract scheme, we construct the model space of functions on the wave spectrum and introduce in that space the model operator. The latter is a matrix Sturm-Liouville operator which is unitarily equivalent to the original.
We prove that the potential of a Sturm--Liouville operator depends analytically and Lipschitz continuously on the spectral data (two spectra or one spectrum and the corresponding norming constants). We treat the class of operators with real-valued distributional potentials in the Sobolev class W^{s-1}_2(0,1), s\in[0,1].
Discovered by M.G.Krein analogy between polinomials orthogonal on the unit circle and generalized eigenfunctions of certain differential systems is used to obtain some new results in the spectral analysis of Sturm-Liouville operators. Some properties of Dirac-type differential systems are studied.
We prove a quantitative version of Gordon's Theorem concerning absence of eigenvalues for Jacobi matrices and Sturm-Liouville operators with complex coefficients.
We consider a recently discovered representation for the general solution of the Sturm-Liouville equation as a spectral parameter power series (SPPS). The coefficients of the power series are given in terms of a particular solution of the Sturm-Liouville equation with the zero spectral parameter. We show that, among other possible applications, this provides a new and efficient numerical method for solving initial value and boundary value problems. Moreover, due to its convenient form the representation lends itself to numerical solution of spectral Sturm-Liouville problems, effectively by cal…
A connection, which shows the dependence of norming constants on boundary conditions, was found using the Gelfand-Levitan method for the solution of inverse Sturm-Liouville problem.
We consider compactly supported perturbations of periodic Sturm-Liouville equations. In this context, one can use the Floquet solutions of the periodic background to define scattering coefficients. We prove that if the reflection coefficient is identically zero, then the operators corresponding to the periodic and perturbed equations, respectively, are unitarily equivalent. In some appendices, we also provide the proofs of several basic estimates, e.g. bounds and asymptotics for the relevant m-functions.
We give a comprehensive treatment of Sturm-Liouville operators with measure-valued coefficients including, a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl-Titchmarsh theory. We avoid previous technical restrictions and, at the same time, extend all results to a larger class of operators. Our operators include classical Sturm-Liouville operators, Lax operators arising in the treatment of the Camassa-Holm equation, Jacobi operators, and Sturm-Liouville operators on time scales as special cases.
We consider standard subordinacy theory for general Sturm--Liouville operators and give criteria when boundedness of solutions implies that no subordinate solutions exist. As applications, we prove a Weidmann-type result for general Sturm--Liouville operators and investigate the absolutely continuous spectrum of radially symmetric quantum trees.
This paper aims to study the q-analogue of the Sturm Liouville problem and to give an asymptotic behaviour at infinity for its solution '. Additionally, we establish an asymptotic expansion of the q-Bessel function $j_α$ for $α>-{1/2}. We are not in situation to claim that our results are new but they have the advantage to show that the method used by Agranovich and Marchenko remain true.
In the paper we construct the wave functional model of a symmetric restriction of the regular Sturm-Liouville operator on an interval. The model is based upon the notion of the wave spectrum and is constructed according to an abstract scheme which was proposed earlier. The result of the construction is a differential operator of the second order on an interval, which differs from the original operator only by a simple transformation.
We consider factorization problem for differential operators on the commutative algebra of densities (defined either algebraically or in terms of an auxiliary extended manifold) introduced in 2004 by Khudaverdian and Voronov in connection with Batalin-Vilkovisky geometry. We consider the case of the line, where unlike the familiar setting (where operators act on functions) there are obstructions for factorization. We analyze these obstructions. In particular, we study the "generalized Sturm-Liouville" operators acting on the algebra of densities on the line. This in a certain sense is in betwe…
The Sturm-Liouville hierarchy of evolution equations was introduced in [Adv. Nonlinear Stud. 11 (2011), 555-591] and includes the Korteweg-de Vries and the Camassa-Holm hierarchies. We discuss some solutions of this hierarchy which are obtained as limits of algebro-geometric solutions. The initial data of our solutions are (generalized) reflectionless Sturm-Liouville potentials [Stoch. Dyn. 8 (2008), 413-449].
We extend relative oscillation theory to the case of Sturm--Liouville operators $H u = r^{-1}(-(pu')'+q u)$ with different $p$'s. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.
This work deals with an inverse problem for the Sturm-Liouville operator with non-separated boundary conditions, one of which linearly depends on a spectral parameter. Uniqueness theorem is proved, solution algorithm is constructed and sufficient conditions for solvability of inverse problem are obtained. As spectral data, we only use the spectrum of one boundary value problem and some sequence of signs.
We establish various uniqueness results for inverse spectral problems of Sturm-Liouville operators with a finite number of discontinuities at interior points at which we impose the usual transmission conditions. We consider both the case of classical Robin and of eigenparameter dependent boundary conditions.
We consider two main inverse Sturm-Liouville problems: the problem of recovery of the potential and the boundary conditions from two spectra or from a spectral density function. A simple method for practical solution of such problems is developed, based on the transmutation operator approach, new Neumann series of Bessel functions representations for solutions and the Gelfand-Levitan equation. The method allows one to reduce the inverse Sturm-Liouville problem directly to a system of linear algebraic equations, such that the potential is recovered from the first element of the solution vector.…
We study the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from their Dirichlet spectra and sequences of the norming constants. For the class of problems under consideration, we give a complete description of the corresponding spectral data, suggest a reconstruction algorithm, and establish uniqueness of reconstruction. The approach is based on connection between spectral problems for energy-dependent Sturm-Liouville equations and for Dirac operators of special form.
We give here some negative results in Sturm-Liouville inverse theory, meaning that we cannot approach any of the potentials with $m+1$ integrable derivatives on $\mathbb{R}^+$ by an $ω$-parametric analytic family better than order of $(ω\lnω)^{-(m+1)}$. Next, we prove an estimation of the eigenvalues and characteristic values of a Sturm-Liouville operator and some properties of the solution of a certain integral equation. This allows us to deduce from [Henkin-Novikova] some positive results about the best reconstruction formula by giving an almost optimal formula of order of $ω^{-m}$.
We consider Sturm-Liouville operators with measure-valued weight and potential, and positive, bounded diffusion coefficient which is bounded away from zero. By means of a local periodicity condition, which can be seen as a quantitative Gordon condition, we prove a bound on eigenvalues for the corresponding operator in $L_p$, for $1\leq p<\infty$. We also explain the sharpness of our quantitative bound, and provide an example for quasiperiodic operators.
In this paper we provide a detailed analysis of the analytic continuation of the spectral zeta function associated with one-dimensional regular Sturm-Liouville problems endowed with self-adjoint separated and coupled boundary conditions. The spectral zeta function is represented in terms of a complex integral and the analytic continuation in the entire complex plane is achieved by using the Liouville-Green (or WKB) asymptotic expansion of the eigenfunctions associated with the problem. The analytically continued expression of the spectral zeta function is then used to compute the functional de…
In this note we provide estimates for the lower bound of the self-adjoint operator associated with the three-coefficient Sturm-Liouville differential expression $$ \frac{1}{r} \left(-\frac{\mathrm d}{\mathrm dx} p \frac{\mathrm d}{\mathrm dx} + q\right) $$ in the weighted $L^2$-Hilbert space $L^2(\mathbb R; rdx)$.
A new approach is proposed to the solution of the quantum mechanical inverse scattering problem at fixed energy. The method relates the fixed energy phase shifts to those arising in an auxiliary Sturm-Liouville problem via the interpolation theory of the Weyl-Titchmarsh m-function. Then a Marchenko equation is solved to obtain the potential.
In this text we describe the spectral nature (pure point or continuous) of a self-similar Sturm-Liouville operator on the line or the half-line. This is motivated by the more general problem of understanding the spectrum of Laplace operators on unbounded finitely ramified self-similar sets. In this context, this furnishes the first example of a description of the spectral nature of the operator in the case where the so-called "Neumann-Dirichlet" eigenfunctions are absent.
Ambarzumyan's theorem for quadratic Sturm-Liouville problem is extended to second order differential systems of dimension d. It is shown that if the spectrum is the same as the spectrum belonging to the zero potential, then the matrix valued functions p and q are both zero
In this paper we consider the Sturm-Liuoville operator in the Hilbert space $L_2$ with the singular complex potential of $W^{-1}_2$ and two-point boundary conditions. For this operator we give sufficient conditions for norm resolvent approximation by the operators of the same class.
The zeros of the eigenfunctions of self-adjoint Sturm-Liouville eigenvalue problems interlace. For these problems interlacing is crucial for completeness. For the complex Sturm-Liouville problem associated with the Schrodinger equation for a non-Hermitian PT-symmetric Hamiltonian, completeness and interlacing of zeros have never been examined. This paper reports a numerical study of the Sturm-Liouville problems for three complex potentials, the large-N limit of a -(ix)^N potential, a quasi-exactly-solvable -x^4 potential, and an ix^3 potential. In all cases the complex zeros of the eigenfuncti…
We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval. Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with two examples for a stationary Schroedinger equation having a generalized Hulthen potential and an eigenvalue problem for a traveling front in the Allen-Cahn equation.
