Singular Sturm-Liouville Operator Research Articles

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.

Asymptotic behavior of the spectral measure density of a singular Sturm-Liouville operator as λ → −∞

Asymptotic behavior of the spectral measure density of a singular Sturm-Liouville operator as λ → −∞

The inverse spectral problem for radial Schrödinger operators on [formula omitted

We consider an inverse spectral problem for a class of singular Sturm–Liouville operators on the unit interval with explicit singularity a ( a + 1 ) / x 2 , a ∈ N , related to the Schrödinger operator with a radially symmetric potential. The purpose of this paper is the global parametrization of potentials by the spectral data λ a and some norming constants κ a . For a = 0 or 1, λ a × κ a is already known to be a global coordinate system on L R 2 ( 0 , 1 ) . Using some transformation operators, we show that this result holds for any non-negative integer a; moreover, we give a description of the isospectral sets.

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.

On inverse problem for singular Sturm-Liouville operator from two spectra

We study an inverse problem with two given spectra for a second-order differential operator with singularity of the type \(\frac{2}{r} + \frac{{\ell (\ell + 1)}}{{r^2 }}\) (here, l is a positive integer or zero) at zero point. It is well known that two spectra {λ n } and {λ n } uniquely determine the potential function q(r) in the singular Sturm-Liouville equation defined on the interval (0, π]. One of the aims of the paper is to prove the generalized degeneracy of the kernel K(r, s). In particular, we obtain a new proof of the Hochstadt theorem concerning the structure of the difference \(\tilde q(r) - q(r)\).

Retraction of: “Dissipative Sturm-Liouville Operators with Nonseparated Boundary Conditions”

A space of boundary values is constructed for the minimal symmetric singular Sturm-Liouville operator in the Hilbert space \(L_{w}^{2}(a,b)(-\infty \le a < b \le \infty )\) with defect index (2,2) (in Weyl’s limit-circle cases at singular points a and b). A description of all maximal dissipative, maximal accretive, selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at a and b. We investigate maximal dissipative operators with, generally speaking, nonseparated boundary conditions. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We also construct a functional model of a dissipative operator and define its characteristic function. We prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operators.

Dissipative Sturm?Liouville Operators in Limit-Point Case

Dissipative singular Sturm–Liouville operators are studied in the Hilbert space Lw2[a,b) (−∞<a<b≤∞), that the extensions of a minimal symmetric operator in Weyl’s limit-point case. We construct a selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh–Weyl function of a selfadjoint operator. Finally, in the case when the Titchmarsh–Weyl function of the selfadjoint operator is a meromorphic in complex plane, we prove theorems on completeness of the system of eigenfunctions and associated functions of the dissipative Sturm–Liouville operators.

Extensions, Dilations and Functional Models of Singular Sturm-Liouville Operators

A space of boundary values is constructed for minimal symmetric singular Sturm-Liouville operator acting in the Hilbert space Lw [a, b), −∞ < a < b ≤ ∞, with deficiency indices (2, 2) (in Weyl’s limit-circle case). A description of all maximal dissipative, maximal accretive, self-adjoint, and other extensions of such a symmetric operator is given in terms of boundary conditions at end points a and b. We investigate maximal dissipative operators with general (coupled or separated) boundary conditions. We construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and determine its characteristic function. We prove the theorem on completeness of the system of eigenfunctions and associated functions of the maximal dissipative operators.

Asymptotic Behavior of the Density of the Spectral Measure of the Singular Sturm–Liouville Operator

holds. Every function α(λ) of this kind is called a spectral function (measure) of the operator L. An upper bound as λ → +∞ for the spectral function α(λ), sinα = 0, in the case of a locally integrable potential q(x) was obtained for the first time in the paper [2], and an asymptotic formula as λ → +∞ was later established in the paper [3]: α(λ) ∼ (2/π)λ sin−2 α if sinα = 0, and α(λ) ∼ (2/3)π−1λ3/2 if sinα = 0. The papers [4–9] deal with the spectral function α(λ) of the Sturm–Liouville problem. The most comprehensive results concerning the asymptotics of the spectral function of the Laplace operator in an arbitrary domain G ⊂ R were obtained in [10, Chap. 1]. The asymptotic behavior of the spectral function for higher-order ordinary differential operators on the entire line and on the half-line was studied in detail in [11]. The asymptotics of α(λ) as λ → +∞ was refined mainly under two kinds of conditions imposed on the potential q : I. q ∈ L(0,+∞); II. there exists an x0 > 0 such that q(x) = 0 for x ≥ x0, ∫ +∞ x0 ( q′|q|−5/2 + |q′′| |q|−3/2 ) dx < ∞, and ∫ +∞ x0 |q|−1/2dx = ∞.

Dissipative Sturm-Liouville Operators with Nonseparated Boundary Conditions

A space of boundary values is constructed for the minimal symmetric singular Sturm-Liouville operator in the Hilbert space \(L_{w}^{2}(a,b)(-\infty \le a < b \le \infty )\) with defect index (2,2) (in Weyl’s limit-circle cases at singular points a and b). A description of all maximal dissipative, maximal accretive, selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at a and b. We investigate maximal dissipative operators with, generally speaking, nonseparated boundary conditions. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We also construct a functional model of a dissipative operator and define its characteristic function. We prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operators.

Buckling of a critically tapered rod: global bifurcation

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Stuart C.A and Vuillaume G 2003Buckling of a critically tapered rod: global bifurcationProc. R. Soc. Lond. A.4591863–1889http://doi.org/10.1098/rspa.2002.1092SectionRestricted accessBuckling of a critically tapered rod: global bifurcation C.A Stuart C.A Stuart Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (; ) Google Scholar Find this author on PubMed Search for more papers by this author and G Vuillaume G Vuillaume Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (; ) Google Scholar Find this author on PubMed Search for more papers by this author C.A Stuart C.A Stuart Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (; ) Google Scholar Find this author on PubMed Search for more papers by this author and G Vuillaume G Vuillaume Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (; ) Google Scholar Find this author on PubMed Search for more papers by this author Published:08 August 2003https://doi.org/10.1098/rspa.2002.1092AbstractThis paper, which can be considered as a continuation of the papers Buckling of a heavy tapered rod (2001, J. Math. Pures Appl. 80, 281–337) and On the spectral theory of a heavy tapered rod (2002, Proc. R. Soc. Edinb. A132, 729–764), is concerned with the study of buckling of a tapered rod. This physical phenomenon leads to the nonlinear eigenvalue problem{A(s)u′(s)}′ + μsinu(s) = 0 for all s ∈ (0,1), <BR></BR>u(1) = lims→0A(s)u′(s) = 0and ∫01A(s)u′(s)2 ds < ∞, where A(s) ∈ C([0,1]) is such that A(s) > 0 for all s > 0 and lims→A(s)/sp = L for some constants p ⩾ 0 and L ∈ (0,∞). We deal with the critical case p = 2 and study the set of all solutions of the problem. In particular, we find the points μ ∈ R+ such that bifurcation occurs at (μ,0).As was shown in the former reference, (μ,u) is a solution of the problem if and only if u ∈ HA and u = μGA(u), where HA denotes the Hilbert space of all admissible configurations and GA : HA → HA is a completely continuous operator. If 0 ⩽ p < 2, GA is Fréchet differentiable and the 2001 paper contains results about global bifurcation derived from the general theory. In the case p = 2, GA is not Fréchet differentiable. We look for the bifurcation points in this critical case, where the abstract theory cannot be invoked. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited byStuart C (2021) Global bifurcation at isolated singular points of the Hadamard derivative, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 379:2191, Online publication date: 22-Feb-2021. Stuart C (2020) Qualitative properties and global bifurcation of solutions for a singular boundary value problem, Electronic Journal of Qualitative Theory of Differential Equations, 10.14232/ejqtde.2020.1.90:90, (1-36) Castro H (2017) The essential spectrum of a singular Sturm-Liouville operator, Mathematische Nachrichten, 10.1002/mana.201600176, 291:4, (593-609), Online publication date: 1-Mar-2018. Stuart C (2016) Bifurcation points of a singular boundary-value problem on (0,1), Journal of Differential Equations, 10.1016/j.jde.2015.12.040, 260:7, (6267-6321), Online publication date: 1-Apr-2016. Stuart C (2015) Bifurcation without Fréchet differentiability at the trivial solution, Mathematical Methods in the Applied Sciences, 10.1002/mma.3409, 38:16, (3444-3463), Online publication date: 15-Nov-2015. Castro H Oscillations in a semi-linear singular Sturm–Liouville equation, Asymptotic Analysis, 10.3233/ASY-151318, 94:3-4, (363-373) Castro H (2014) Bifurcation analysis of a singular nonlinear Sturm–Liouville equation, Communications in Contemporary Mathematics, 10.1142/S0219199714500126, 16:05, (1450012), Online publication date: 1-Oct-2014. Ram Y (2014) Nonlinear eigenvalue problems of the elastica, Mechanical Systems and Signal Processing, 10.1016/j.ymssp.2013.12.006, 45:2, (408-423), Online publication date: 1-Apr-2014. Evéquoz G and Stuart C On differentiability and bifurcation Advances in Mathematical Economics, 10.1007/4-431-30899-7_6, (155-184) Stuart C and Vuillaume G (2004) Buckling of a critically tapered rod: properties of some global branches of solutions, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 460:2051, (3261-3282), Online publication date: 8-Nov-2004. This Issue08 August 2003Volume 459Issue 2036 Article InformationDOI:https://doi.org/10.1098/rspa.2002.1092Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/08/2003Published in print08/08/2003 License: Citations and impact Keywordsessential spectrumEuler elasticabifurcation

An Application of Gesztesy–Simon–Teschl Oscillation Theory to a Problem in Differential Geometry

The approach to oscillation theory developed by Gesztesy, Simon, and Teschl produces a sharp version for the oscillation theorem for singular Sturm–Liouville operators. In the present note, an example from the stability theory of complete minimal surfaces is given in which this refinement plays a decisive role.

Spectral inclusion and spectral exactness for singular non-self-adjoint Sturm-Liouville problems

We consider the effect of regularization by interval truncation on the spectrum of a singular non-selfadjoint Sturm-Liouville operator. We present results on spectral inclusion and spectral exactness for the cases where the singularity is in Sims Case II or Sims Case III. For Sims Case I we present a test for spectral inexactness, which can be used to detect when the interval truncation process is generating spurious eigenvalues. Numerical results illustrate the effectiveness of this test.

Inverse problem of acoustic scattering for centrally symmetric finite objects in two-dimensional space

Scattering theory for the wave equation in two-dimensional space, perturbed by a finite function of a radial variable, integrable everywhere except, perhaps, the origin of coordinates, is considered from the point of view of the LaxPhillips scheme. The compression operator, related to the corresponding scattering problem, is considered. It is shown that this compression has one-dimensional defect subspaces, and its characteristic operator-function is a meromorphic function, whose zeros and poles coincide, respectively, with the corresponding values of a dissipative operator and its adjoint. The solution of the inverse scattering problem is obtained by reducing it to the inverse problem with two spectra for the singular self-adjoint Sturm-Liouville operator.

Inverse spectral theory for a singular Sturm-Liouville operator on [0, 1

xeR3 acting on functions in P( 1x1< 1) vanishing on 1x1 = 1. We set x = 1x1 and assume q(x) is a real-valued function in L*( [0, 11). The closure H of I? is a self-adjoint operator with compact resolvent. Hence H has pure point spectrum without finite accumulation points. Moreover, using the mapping U(X) + xu(x) and expansion in spherical harmonics, one shows that H is unitarily equivalent to a direct sum of Sturm-Liouville operators acting on L*[O, 11, cf. [ 11. The first operator in this sum is

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

SELF-ADJOINTNESS IN THE WEYL LIMIT CASES. THE ROLE OF BOUNDARY CONDITIONS

ABSTRACT The role of boundary conditions in assuring self-adjointness for the singular second order Sturm-Liouville operator -yn + qy is discussed, both in the limit circle and limit point cases.

Heat equation asymptotics for singular sturm-Liouville operators

Soit a∈C ∞ (R) a valeur reelle et l'expression differentielle #7B-T:=-d x 2 +a(x)/(x 2 )(1−x) 2 =:−d x 2 +a(x) dans I:=(0,1). Si a(i)≥−1/4, i=0,1, l'extension de Friedrichs T de #7B-T/C 0 ∞ (I) existe dans L 2 (I) et a un spectre ponctuel pur. On etudie le developpement asymptotique de tr exp(−sT) quand s→0+

Norm estimates for generalized translation operators associated with a singular differential operator

Delsarte's approach via the solution of a Cauchy problem is used to estimate generalized translation operators associated with a singular Sturm-Liouville differential operator of the form D q, x α ,=( d 2/ dx 2) + ((2 α + 1)/ x)( d/ dx)- q( x), a≥ 1 2 , 0<×<∞. Such estimates are fundamental for establishing a convolution structure. As a general purpose, the relation between translations belonging to different potential functions q are studied. This leads to norm estimates for classes of “related” translation operators mainly under hypotheses on the Riemann function associated with an explicitly given member of the class. Specifically, our aim is to investigate perturbations of the Hankel translation as well as of the Laguerre translation, the underlying Sturm-Liouville equations of which are prototypes of singular equations on the positive half-axis with continuous and discrete spectra, respectively.

Symmetric operators with singular spectral functions

In a previous article [3] it was shown that if A is a closed symmetric operator with deficiency indices (1, 1) in the Hilbert space H and if ,Ao is a point of regular type for A, then there is a neighborhood of /Ao in which every minimal selfadjoint dilation A+ of A has spectral multiplicity not more than 1. This shows that in Theorem 14 of Coddington [2] and Theorem 5.2 of McKelvey [6] it is the hypothesis that every point is of regular type which determines the spectral multiplicity of the dilation; the hypothesis that the contraction F(X) is continuous down to the real axis and has norm less than 1 there serves to make the spectral function absolutely continuous. (See Remark 2.) The procedure used in [31 depended on the fact that if .o is a point of regular type for A, then there exists a selfadjoint extension A o of A for which ,Ao is in the resolvent set. This means that for different points IO one might have to choose different operators Ao. In the present note this necessity is eliminated. It is shown that if A has a selfadjoint extension Ao with a pure point spectrum with no finite limit points, then for each minimal selfadjoint extension or dilation A+ of A there is defined on the real axis a nondecreasing function p(a) (which depends solely on A+ and on the single selfadjoint extension Ao) such that A+ is unitarily equivalent to the multiplication operator in L,. Kre1n [5] has shown that if A is simple, then there exists a selfadjoint extension A0 of the type described above if and only if every point is of regular type for A. (See Remark 1 also.) In the case that A is a singular Sturm-Liouville operator, the results of the paper are extended to the case that there exists a selfadjoint extension Ao with a singular spectral function. For terminology used we refer the reader to [3] and to Achieser and Glasmann [1]. We note first of all the following lemma due to Straus [8].