Singular Sturm-Liouville Operator Research Articles

Fractional singular Sturm-Liouville problems on the half-line

In this paper, we consider two types of singular fractional Sturm-Liouville operators. One comprises the composition of left-sided Caputo and left-sided Riemann-Liouville derivatives of order alpha in(0,1). The other one is the composition of left-sided Riemann-Liouville and right-sided Caputo derivatives. The reality of the corresponding eigenvalues and the orthogonality of the eigenfunctions are proved. Furthermore, we formulate the fractional Laguerre Strum-Liouville problems and derive the explicit eigenfunctions as the non-polynomial functions related to Laguerre polynomials. Finally, we introduce the generalized Laguerre transform and employ it to solve the unbounded space-fractional diffusion equations.

The essential spectrum of a singular Sturm–Liouville operator

AbstractIn this paper we study the essential spectrum of the operator urn:x-wiley:0025584X:media:mana201600176:mana201600176-math-0001where is a positive absolutely continuous function on (0, 1) that resembles for some . We prove that the essential spectrum of coincides with the essential spectrum of the operator .

Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators

Abstract In this study we construct a space of boundary values of the minimal symmetric discrete Sturm-Liouville (or second-order difference) operators with defect index (1, 1) (in limit-circle case at ±∞ and limit-point case at ∓∞), acting in the Hilbert space ℓ ϱ 2 ( Z ) ℓ ϱ 2 ( Z ) ( Z := { 0 , ± 1 , ± 2 , … } ) $\ell_{\varrho}^{2}(\mathbb{Z}) (\mathbb{Z} :=\{0,\pm 1,\pm 2,\dots\})$ . Such a description of all maximal dissipative, maximal accumulative and self-adjoint extensions is given in terms of boundary conditions at ± ∞. After constructing the space of the boundary values, we investigate two classes of maximal dissipative operators. This investigation is done with the help of the boundary conditions, called “dissipative at −∞” and “dissipative at ∞”. In each of these cases we construct a self-adjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations. These representations allow us to determine the scattering matrix of dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the Weyl-Titchmarsh function of the self-adjoint operator. Finally, we prove a theorem on completeness of the system of eigenvectors and associated vectors (or root vectors) of the maximal dissipative operators.

Inverse Spectral Theory for a Singular Sturm Liouville Operator with Coulomb Potential

We consider the inverse spectral problem for a singular Sturm-Liouville operator with Coulomb potential. In this paper, we give an asymptotic formula and some properties for this problem by using methods of Trubowitz and Poschel.

Asymptotic behavior of eigenvalues of hydrogen atom equation

We consider a spectral problem for a class of singular Sturm-Liouville operators on the unit interval with explicit singularity $2/x$ - $2/x^{2}$ , related to the Schrodinger operator with radially symmetric potential. In particular, we give the asymptotic behavior of the eigenvalues of the hydrogen atom equation.

An Invariance Principle to Ferrari–Spohn Diffusions

We prove an invariance principle for a class of tilted (1+1)-dimensional SOS models or, equivalently, for a class of tilted random walk bridges in Z_+. The limiting objects are stationary reversible ergodic diffusions with drifts given by the logarithmic derivatives of the ground states of associated singular Sturm-Liouville operators. In the case of a linear area tilt, we recover the Ferrari-Spohn diffusion with log-Airy drift, which was derived by Ferrari and Spohn in the context of Brownian motions conditioned to stay above circular and parabolic barriers.

Comparison and oscillation theorems for singular Sturm-Liouville operators

We prove analogues of the classical Sturm comparison and oscillation theorems for Sturm-Liouville operators on a finite interval with real-valued distributional potentials.

Fractional singular Sturm-Liouville operator for Coulomb potential

In this paper, we define a fractional singular Sturm-Liouville operator having Coulomb potential of type . Our main issue is to investigate the spectral properties for the operator. Furthermore, we prove new results according to the fractional singular Sturm-Liouville problem. MSC:26A33, 34A08.

On Finite Rank Perturbations of Selfadjoint Operators in Krein Spaces and Eigenvalues in Spectral Gaps

It is shown that the finiteness of eigenvalues in a spectral gap of a definitizable or locally definitizable selfadjoint operator in a Krein space is preserved under finite rank perturbations. This results is applied to a class of singular Sturm–Liouville operators with an indefinite weight function.

Trace of a difference of singular Sturm-Liouville operators with a potential containing Dirac δ-functions

In L2[0,+∞) we consider the Sturm-Liouville operator generated by the expression involving delta functions and evaluate the trace of the difference between two operators of this kind which differ by a multiplication operator by a locally summable function on the semiaxis.

On the calculation of the finite Hankel transform eigenfunctions

In this work, we discuss the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform, important for numerous applications. Due to the very special behavior of the Hankel transform eigenfunctions, their direct numerical calculation often causes an essential loss of accuracy.Here, we present several simple, efficient and robust numerical techniques to compute Hankel transform eigenfunctions via the associated singular self-adjoint Sturm-Liouville operator. The properties of the proposed approaches are compared and illustrated by means of numerical experiments.

NONSELFADJOINT SINGULAR STURM-LIOUVILLE OPERATORS IN LIMIT-CIRCLE CASE

In this paper, we study the maximal dissipative singular Sturm-Liouville operators (in Weyl's limit-circle case at singular point $b$) acting in the Hilbert space $L_{w}^{2}\left[ a,b\right)$ ($-\infty \lt a\lt b\leq \infty$ ). In fact, we consider all extensions of a minimal symmetric operator and we investigate two classes of maximal dissipative operators with separated boundary conditions, called 'dissipative at $a$' and 'dissipative at $b$'. In both cases, we construct a selfadjoint dilation of the maximal dissipative operator and determine its incoming and outgoing spectral representations. This representations make it possible to determine the scattering matrix ofthe dilation in terms of the Titchmarsh-Weyl function of a selfadjoint Sturm-Liouville operator. We also construct a functional model of the maximal dissipative operator and determine its characteristic function in terms of the scattering matrix of the dilation (or of the Titchmarsh-Weyl function). Finally we prove theorems on the completeness of the eigenfunctions and associated functions of the maximal dissipative Sturm-Liouville operators.

Half inverse problem for singular Sturm-Liouville operators with discontinuity conditions inside an interval

In this study, half inverse problem for singular Sturm-Liouville operator is considered. It is shown by Hochstadt and Lieberman's method that if the potential function q(x) prescribed on interval ((π/2),π), then a single spectrum sufficies to determine q(x) on the whole interval (0,π).

Singular Sturm-Liouville operators with distribution potential on spaces of vector functions

Singular Sturm-Liouville operators with distribution potential on spaces of vector functions

Eigenvalue estimates for singular left-definite Sturm–Liouville operators

The spectral properties of a singular left-definite Sturm-Liouville operator JA are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart A which is obtained by substituting the indefinite weight function by its absolute value. The spectrum of the J -selfadjoint operator JA is real and it follows that an interval .a; b/ � R C is a gap in the essential spectrum of A if and only if both intervals .� b; � a/ and .a; b/ are gaps in the essential spectrum of the J -selfadjoint operator JA. As one of the main results it is shown that the number of eigenvalues of JA in .� b; � a/ ( .a; b/ differs at most by three from the number of eigenvalues of A in the gap .a; b/; as a byproduct results on the accumulation of eigenvalues of singular left-definite Sturm-Liouville operators are obtained. Furthermore, left-definite problems with symmetric and periodic coefficients are treated, and several examples are included to illustrate the general results.

Regular singular Sturm–Liouville operators and their zeta-determinants

We consider Sturm–Liouville operators on the line segment [ 0 , 1 ] with general regular singular potentials and separated boundary conditions. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski-determinant of a fundamental system of solutions adapted to the boundary conditions. This generalizes the earlier work of the first author, treating general regular singular potentials but only the Dirichlet boundary conditions at the singular end, and the recent results by Kirsten–Loya–Park for general separated boundary conditions but only special regular singular potentials.

Definitizability of a Class of J‐Selfadjoint Operators with Applications

AbstractWe formulate an abstract result concerning the definitizability of J‐selfadjoint operators which, roughly speaking, differ by at most finitely many dimensions from the orthogonal sum of a J‐selfadjoint operator with finitely many negative squares and a semibounded selfadjoint operator in a Hilbert space. The general perturbation result is applied to a class of singular Sturm–Liouville operators with indefinite weight functions. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Accumulation of complex eigenvalues of indefinite Sturm–Liouville operators

Spectral properties of singular Sturm–Liouville operators of the form with the indefinite weight x ↦ sgn(x) on are studied. For a class of potentials with lim|x|→∞V(x) = 0 the accumulation of complex and real eigenvalues of A to zero is investigated and explicit eigenvalue problems are solved numerically.

Scattering matrices and Weyl functions

For a scattering system {AΘ, A0} consisting of self-adjoint extensions AΘ and A0 of a symmetric operator A with finite deficiency indices, the scattering matrix {SΘ(λ)} and a spectral shift function ξΘ are calculated in terms of the Weyl function associated with a boundary triplet for A*, and a simple proof of the Krein–Birman formula is given. The results are applied to singular Sturm–Liouville operators with scalar and matrix potentials, to Dirac operators and to Schrödinger operators with point interactions.

On the approximation of isolated eigenvalues of ordinary differential operators

We extend a result of Stolz and Weidmann on the approximation of isolated eigenvalues of singular Sturm-Liouville and Dirac operators by the eigenvalues of regular operators.