Sturm-Liouville Differential Expressions Research Articles

Relative oscillation theory and essential spectra of Sturm–Liouville operators

We develop relative oscillation theory for general Sturm–Liouville differential expressions of the form1r(−ddxpddx+q) and prove perturbation results and invariance of essential spectra in terms of the real coefficients p, q, r. The novelty here is that we also allow perturbations of the weight function r in which case the unperturbed and the perturbed operator act in different Hilbert spaces.

Strict domain monotonicity of the principal eigenvalue and a characterization of lower boundedness for the Friedrichs extension of four-coefficient Sturm–Liouville operators

Using the variational characterization of the principal (i.e., smallest) eigenvalue below the essential spectrum of a lower semibounded self-adjoint operator, we prove strict domain monotonicity (with respect to changing the finite interval length) of the principal eigenvalue of the Friedrichs extension \( T_F\) of the minimal operator for regular four-coefficient Sturm–Liouville differential expressions. In the more general singular context, these four-coefficient differential expressions act according to $$\tau f= \frac{1}{r}(-(f^{[1]})'+sf^{[1]}+qf)\, {\rm with}\, f^{[1]}=p[f'+sf]\, {\rm on}\, (a,b) \subseteq \mathbb{R}, $$ where the coefficients p, q, r, s are real-valued and Lebesgue measurable on (a, b), with p > 0, r > 0 a.e. on (a, b), and \( p^{-1},q,r,s \in L^{1}_{loc}((a,b);dx)\), and \( f\) is supposed to satisfy $$f\in AC_{loc}((a,b)),p[f'+sf] \in AC_{loc}((a,b)). $$ This setup is sufficiently general so that \(\tau \) permits certain distributional potential coefficients q, including potentials in \(H^{-1}_{loc}((a,b)) \).As a consequence of the strict domain monotonicity of the principal eigenvalue of the Friedrichs extension in the regular case, and on the basis of oscillation theory in the singular context, in our main result, we characterize all lower bounds of \(T_F \) as those \(\lambda \in \mathbb{R} \) for which the differential equation \(\tau u =\lambda u \) has a strictly positive solution u > 0 on (a, b).

One-dimensional Schrödinger operators with [formula omitted]-interactions on Cantor-type sets

We introduce a novel approach for defining a δ′-interaction on a subset of the real line of Lebesgue measure zero which is based on Sturm–Liouville differential expression with measure coefficients. This enables us to establish basic spectral properties (e.g., self-adjointness, lower semiboundedness and spectral asymptotics) of Hamiltonians with δ′-interactions concentrated on sets of complicated structures.

The similarity problem for indefinite Sturm–Liouville operators and the HELP inequality

We study two problems. The first one is the similarity problem for the indefinite Sturm–Liouville operator A=−(sgnx)dwdxdrdx acting in Lw2(−b,b). It is assumed that w,r∈Lloc1(−b,b) are even and positive a.e. on (−b,b).The second object is the so-called HELP inequality (∫0b1r̃|f′|dx)2≤K2∫0b|f|2w̃dx∫0b|1w̃(1r̃f′)′|2w̃dx, where the coefficients w̃,r̃∈Lloc1[0,b) are positive a.e. on (0,b).Both problems are well understood when the corresponding Sturm–Liouville differential expression is regular. The main objective of the present paper is to give criteria for both the validity of the HELP inequality and the similarity to a self-adjoint operator in the singular case. Namely, we establish new criteria formulated in terms of the behavior of the corresponding Weyl–Titchmarsh m-functions at 0 and at ∞. As a biproduct of this result we show that both problems are closely connected. Namely, the operator A is similar to a self-adjoint one precisely if the HELP inequality with w̃=r and r̃=w is valid.Next we characterize the behavior of m-functions in terms of coefficients and then these results enable us to reformulate the obtained criteria in terms of coefficients. Finally, we apply these results for the study of the two-way diffusion equation, also known as the time-independent Fokker–Planck equation.

Dissipative non-self-adjoint Sturm–Liouville operators and completeness of their eigenfunctions

In this paper, non-self-adjoint Sturm–Liouville operators in Weyl’s limit-circle case are studied. We first determine all the non-self-adjoint boundary conditions yielding dissipative operators for each allowed Sturm–Liouville differential expression. Then, using the characteristic determinant, the completeness of the system of eigenfunctions and associated functions for these dissipative operators is proved.

On the Product of Self-Adjoint Sturm-Liouville Differential Operators in Direct Sum Spaces

In this paper, the second-order symmetric Sturm-Liouville differential expressions $\tau _1,\tau _2,\ldots,\tau _n$ with real coefficients on any finite number of intervals are studied in the setting of the direct sum of the $L_w^2$-spaces of functions defined on each of the separate intervals. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, it is an exact parallel of that in the regular case. This characterization is an extension of those obtained in [6]}, [7], [8], [9], [12], [14] and [15].

The Evans function and the Weyl-Titchmarsh function

We describe relations between the Evans function, a modern tool in the study of stability of traveling waves and other patterns for PDEs, and the classical Weyl-Titchmarsh function for singular Sturm-Liouville differential expressions and for matrix Hamiltonian systems. Also, for the scalar Schrodinger equation, we discuss a related issue of approximating eigenvalue problems on the whole line by that on finite segments.

Asymptotic behavior of the solutions and nodal points of Sturm-Liouville differential expressions

We obtain asymptotic formulas for the values of second-order differential operators with the coefficient of bounded variation depending on the spectral parameter. We give a counterexample showing that the requirement of boundedness for the variation is essential for the preservation of the error of the so-obtained asymptotic formulas.

On the spectral theory of singular indefinite Sturm–Liouville operators

We consider a singular Sturm–Liouville differential expression with an indefinite weight function and we show that the corresponding self-adjoint differential operator in a Krein space locally has the same spectral properties as a definitizable operator.

The Completeness of Eigenfunctions of Perturbation Connected with Sturm-Liouville Operators

In this paper, non-self-adjoint Sturm-Liuville operators in Weyl’s limit-circle case are studied. We first determine all the non-self-adjoint boundary conditions yielding dissipative operators for each allowed Sturm-Liouville differential expression. Then, using the characteristic determinant, we prove the completeness of the system of eigenfunctions and associated functions for these dissipative operators.

On the product of self-adjoint Sturm-Liouville differential operators in direct sum spaces

In this paper, the second-order symmetric Sturm-Liouville differential expressions $ \tau_1,\tau_2, \ldots, \tau_n $, with real coefficients on any finite number of intervals are studied in the setting of the direct sum of the $ L_w^2 $-spaces of functions defined on each of the separate intervals. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, it is an exact parallel of that in the regular case. This characterization is an extension of those obtained in [6], [7], [8], [9], [12], [14] and [15].

On the boundary conditions for products of Sturm-Liouville differential operators

In this paper, the second-order symmetric Sturm-Liouville differential expressions $ \tau_1, \tau_2, \ldots, \tau_n $ with real coefficients are considered on the interval $ I = (a,b) $, $ - \infty \le a < b \le \infty $. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximan domain of the product operators, and is an exact parallel of the regular case. This characterization is an extension of those obtained in [6], [8], [11-12], [14] and [15].

SEPARATION PROBLEM FOR STURM-LIOUVILLE EQUATION WITH OPERATOR COEFFICIENT

Let H be a separable Hilbert Space. Denote by H1 = L2(a,b; H) the set of function defned on the interval a < c < b (¾¥ a < c < b £ ¥)whose values belong to H strongly measurable [12] and satisfying the conditionZ b a ||f(x)||2 Hdx < ? If the inner product of function ¦(c) and g(c) belonging to H1 is defined by (f, g)1 = Z b a (f(x), g(x))Hdx then H1 forms a separable Hilbert space. We study separation problem for the operator formed by ¾ y"+ Q (c) y Sturm-Liouville differential expression in L2(¾ ¥, ¥; H) space has been proved where Q (c) in an operator which transforms at H in value of c,,self-adjoint, lower bounded and its inverse is complete continous.

Dissipative Operators Generated by the Sturm–Liouville Differential Expression in the Weyl Limit Circle Case

In this paper, using Livšic's theorem, we investigate the problem of completeness of the system of eigenfunctions and associated functions of dissipative operators generated by the Sturm–Liouville differential expression on the semi-axis in Weyl's limit-circle case.

The Determinants of Perturbation Connected with a Dissipative Sturm-Liouville Operator

In this paper, the determinants of perturbation connected with a dissipative operator generated by the Sturm-Liouville differential expression on the half-line in the Weyl′s limit-circle case are investigated. An application is given to the problem of completeness of the system of eigenvectors and associated vectors of this operator.

On the conditions of solvability and the resolvent of a second-order deferential boundary operator in a space of vector functions

A boundary differential operator generated by the Sturm-Liouville differential expression with bounded operator potential and nonlocal boundary conditions is considered. The conditions for a considered operator to be a Fredholm and solvable operator are established and its resolvent is constructed.

On a Conjecture of J. Weidmann

Consider the Sturm-Liouville differential expression l(y) = −y″ +q(x)y on an interval (a,b) and assume that l is in the limit point case at b. Fix c ∈(a,b) and let L, Lb be self-adjoint realizations of l in ℒ2(a,b), ℒ2(c,b) respectively. If Lb has purely absolutely continuous spectrum in an interval J and if the spectral function ρb of Lb satisfies some mild growth conditions then the spectrum of L in J is shown to be purely absolutely continuous, too. Our result confirms a conjecture of J. Weidmann (1982). It had been shown by del Rio Castillo (1988) that in Weidmann’s original formulation this conjecture is false.

Spectral representation of a hyperbolic evolutionary process connected with a Sturm-Liouville differential expression

Spectral representation of a hyperbolic evolutionary process connected with a Sturm-Liouville differential expression

Characterizations of the Friedrichs Extensions of Singular Sturm–Liouville Expressions

Previous article Next article Characterizations of the Friedrichs Extensions of Singular Sturm–Liouville ExpressionsHans G. Kaper, Man Kam Kwong, and Anton ZettlHans G. Kaper, Man Kam Kwong, and Anton Zettlhttps://doi.org/10.1137/0517056PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA method is presented to characterize selfadjoint realizations of a singular Sturm–Liouville differential expression on a finite interval, where the singularities are of limit-circle type.[1] M. A. Naimark, Linear differential operators. Part I: Elementary theory of linear differential operators, Frederick Ungar Publishing Co., New York, 1967xiii+144 35:6885 M. A. Naimark, Linear differential operators. Part II: Linear differential operators in Hilbert space, With additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt, Frederick Ungar Publishing Co., New York, 1968xv+352 41:7485 Google Scholar[2] N. I. Akhiezer and , I. M. Glazman, Theory of linear operators in Hilbert space. Vol. I, Translated from the Russian by Merlynd Nestell, Frederick Ungar Publishing Co., New York, 1961xi+147 41:9015a N. I. Akhiezer and , I. M. Glazman, Theory of linear operators in Hilbert space. Vol. II, Translated from the Russian by Merlynd Nestell, Frederick Ungar Publishing Co., New York, 1963v+218 41:9015b Google Scholar[3] H. Weyl, Üher gewöhnliche Differentialgleichungen mit Singulärituten and die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann, 68 (1910), 220–269 CrossrefGoogle Scholar[4] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Vol. 2, Oxford Univ. Press, Cambridge, 1962 CrossrefGoogle ScholarKeywordsSturm–Liouville differential operatorssingularities of limit-circle typeselfadjoint realizationsFriedrichs extension Previous article Next article FiguresRelatedReferencesCited ByDetails Friedrichs extensions of a class of singular Hamiltonian systemsJournal of Differential Equations, Vol. 293 | 1 Aug 2021 Cross Ref On self-adjoint boundary conditions for singular Sturm–Liouville operators bounded from belowJournal of Differential Equations, Vol. 269, No. 9 | 1 Oct 2020 Cross Ref Friedrichs extensions for singular Hamiltonian operators with intermediate deficiency indicesJournal of Mathematical Analysis and Applications, Vol. 461, No. 2 | 1 May 2018 Cross Ref On Properties of the Legendre Differential ExpressionResults in Mathematics, Vol. 42, No. 1-2 | 16 May 2013 Cross Ref The Friedrichs Extension of Singular Differential OperatorsJournal of Differential Equations, Vol. 160, No. 2 | 1 Jan 2000 Cross Ref Density, spectral theory and homoclinics for singular Sturm-Liouville systemsJournal of Computational and Applied Mathematics, Vol. 52, No. 1-3 | 1 Jul 1994 Cross Ref Singular Second-Order Operators: The Maximal and Minimal Operators, and Selfadjoint Operators in BetweenMojdeh Hajmirzaahmad and Allan M. KrallSIAM Review, Vol. 34, No. 4 | 2 August 2006AbstractPDF (1871 KB)Eigenvalue and eigenfunction computations for Sturm-Liouville problemsACM Transactions on Mathematical Software, Vol. 17, No. 4 | 1 Dec 1991 Cross Ref Volume 17, Issue 4| 1986SIAM Journal on Mathematical Analysis761-1035 History Submitted:17 August 1984Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsKeywordsSturm–Liouville differential operatorssingularities of limit-circle typeselfadjoint realizationsFriedrichs extensionMSC codes34B2547E05PDF Download Article & Publication DataArticle DOI:10.1137/0517056Article page range:pp. 772-777ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

Full- and partial-range eigenfunction expansions for Sturm-Liouville problems with indefinite weights

SynopsisThis article is concerned with eigenvalue problems of the form Au = λTu in a Hilbert space H, where Ais a selfadjoint positive operator generated by a second-order Sturm-Liouville differential expression and T a selfadjoint indefinite multiplicative operator which is one-to-one. Emphasis is on the full-range and partial-range expansionproperties of the eigenfunctions.