A spectral theory for a λ-rational Sturm-Liouville problem

V. Adamjan, Heinz Langer, M. Langer

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

We consider the regular Sturm-Liouville problem y″−py+(λ+q/(u−λ)) y=0, which contains the eigenvalue parameter rationally. Under certain assumptions on p, q, and u it is shown that the spectrum of the problem consists of a continuous component (the range of the function u), discrete eigenvalues, and possibly a finite number of embedded eigenvalues. In the considered situation the continuous spectrum is absolutely continuous, and explicit formulas for the spectral density and the corresponding Fourier transform are given.
LanguageEnglish
Pages315-345
Number of pages31
JournalJournal of Differential Equations
Volume171
Issue number2
DOIs
Publication statusPublished - Feb 2001

Fingerprint

Sturm-Liouville Problem
Spectral Theory
Eigenvalue
Continuous Spectrum
Spectral Density
Absolutely Continuous
Explicit Formula
Fourier transform
Range of data

Keywords

  • nonlinear eigenvalue problem
  • spectral density
  • block operator matrix
  • numerical mathematics
  • differential equations

Cite this

Adamjan, V. ; Langer, Heinz ; Langer, M. / A spectral theory for a λ-rational Sturm-Liouville problem. In: Journal of Differential Equations. 2001 ; Vol. 171, No. 2. pp. 315-345.
@article{c2794c725b1742d3807c5652504b09be,
title = "A spectral theory for a λ-rational Sturm-Liouville problem",
abstract = "We consider the regular Sturm-Liouville problem y″−py+(λ+q/(u−λ)) y=0, which contains the eigenvalue parameter rationally. Under certain assumptions on p, q, and u it is shown that the spectrum of the problem consists of a continuous component (the range of the function u), discrete eigenvalues, and possibly a finite number of embedded eigenvalues. In the considered situation the continuous spectrum is absolutely continuous, and explicit formulas for the spectral density and the corresponding Fourier transform are given.",
keywords = "nonlinear eigenvalue problem, spectral density, block operator matrix, numerical mathematics, differential equations",
author = "V. Adamjan and Heinz Langer and M. Langer",
year = "2001",
month = "2",
doi = "10.1006/jdeq.2000.3841",
language = "English",
volume = "171",
pages = "315--345",
journal = "Journal of Differential Equations",
issn = "0022-0396",
number = "2",

}

A spectral theory for a λ-rational Sturm-Liouville problem. / Adamjan, V.; Langer, Heinz; Langer, M.

In: Journal of Differential Equations, Vol. 171, No. 2, 02.2001, p. 315-345.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A spectral theory for a λ-rational Sturm-Liouville problem

AU - Adamjan, V.

AU - Langer, Heinz

AU - Langer, M.

PY - 2001/2

Y1 - 2001/2

N2 - We consider the regular Sturm-Liouville problem y″−py+(λ+q/(u−λ)) y=0, which contains the eigenvalue parameter rationally. Under certain assumptions on p, q, and u it is shown that the spectrum of the problem consists of a continuous component (the range of the function u), discrete eigenvalues, and possibly a finite number of embedded eigenvalues. In the considered situation the continuous spectrum is absolutely continuous, and explicit formulas for the spectral density and the corresponding Fourier transform are given.

AB - We consider the regular Sturm-Liouville problem y″−py+(λ+q/(u−λ)) y=0, which contains the eigenvalue parameter rationally. Under certain assumptions on p, q, and u it is shown that the spectrum of the problem consists of a continuous component (the range of the function u), discrete eigenvalues, and possibly a finite number of embedded eigenvalues. In the considered situation the continuous spectrum is absolutely continuous, and explicit formulas for the spectral density and the corresponding Fourier transform are given.

KW - nonlinear eigenvalue problem

KW - spectral density

KW - block operator matrix

KW - numerical mathematics

KW - differential equations

U2 - 10.1006/jdeq.2000.3841

DO - 10.1006/jdeq.2000.3841

M3 - Article

VL - 171

SP - 315

EP - 345

JO - Journal of Differential Equations

T2 - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 2

ER -