Spectral problems of non-self-adjoint q-Sturm-Liouville operators in limit-point case

SynopsisIn this paper, the formally J-symmetric Sturm-Liouville operator with complex-valued coefficients is considered and a generalisation of the Weyl limit-point, limit-circle dichotomy is sought by means of m (λ )-functions. These functions are then used to give an explicit description of all the associated J-selfadjoint operators with separated boundary conditions in the limit-circle case. A formulation of the eigenvalues of these operators, and a characterisation of which extensions are non-well-posed, are also found. Finally, the limit-point case is studied, mainly by means of an example.

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Partial inverse problems for Sturm–Liouville operators on trees

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210516000482 ◽

2017 ◽

Vol 147 (5) ◽

pp. 917-933 ◽

Cited By ~ 9

Author(s):

Natalia Bondarenko ◽

Chung-Tsun Shieh

Keyword(s):

Inverse Problems ◽

A Priori ◽

Potential Functions ◽

Inverse Spectral Problems ◽

Spectral Sets ◽

Spectral Problems ◽

Partial Inverse ◽

Method Of Spectral Mappings ◽

Sturm Liouville ◽

Liouville Operators

In this paper, inverse spectral problems for Sturm–Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b – 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.

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q-Titchmarsh-Weyl theory: series expansion

Nagoya Mathematical Journal ◽

10.1017/s002776300001045x ◽

2012 ◽

Vol 205 ◽

pp. 67-118

Author(s):

M. H. Annaby ◽

Z. S. Mansour ◽

I. A. Soliman

Keyword(s):

Limit Point ◽

Eigenfunction Expansion ◽

Bessel Functions ◽

Sufficient Conditions ◽

Limit Circle ◽

Worked Example ◽

Weyl Theory ◽

Sturm Liouville ◽

Limit Point Case ◽

Theory Series

AbstractWe establish aq-Titchmarsh-Weyl theory for singularq-Sturm-Liouville problems. We defineq-limit-point andq-limit circle singularities, and we give sufficient conditions which guarantee that the singular point is in a limit-point case. The resolvent is constructed in terms of Green’s function of the problem. We derive the eigenfunction expansion in its series form. A detailed worked example involving Jacksonq-Bessel functions is given. This example leads to the completeness of a wide class ofq-cylindrical functions.

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INVERSE SPECTRAL PROBLEMS FOR STURM–LIOUVILLE OPERATORS WITH SINGULAR POTENTIALS. IV. POTENTIALS IN THE SOBOLEV SPACE SCALE

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091504000859 ◽

2006 ◽

Vol 49 (2) ◽

pp. 309-329 ◽

Cited By ~ 20

Author(s):

Rostyslav O. Hryniv ◽

Yaroslav V. Mykytyuk

Keyword(s):

Sobolev Space ◽

Spectral Data ◽

Sufficient Conditions ◽

Inverse Spectral Problems ◽

Singular Potentials ◽

Spectral Problems ◽

Sturm Liouville ◽

Inverse Spectral ◽

Necessary And Sufficient ◽

Liouville Operators

AbstractWe solve the inverse spectral problems for the class of Sturm–Liouville operators with singular real-valued potentials from the Sobolev space $W^{s-1}_2(0,1)$, $s\in[0,1]$. The potential is recovered from two spectra or from one spectrum and the norming constants. Necessary and sufficient conditions for the spectral data to correspond to a potential in $W^{s-1}_2(0,1)$ are established.

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