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Titchmarsh-Weyl M-function asymptotics and some results in the inverse spectral theory for vector-valued Sturm-Liouville equations and a certain higher order ordinary differential equation

Andersson, Erik LU (2002)
Abstract (Swedish)
Popular Abstract in Swedish

Denna avhandling utgöres av två separata rapporter, där studieobjektet för den första är en vektorvärd Sturm-Liouville ekvation på formen -(PU')'+QU=zU, xin[0,b), med P^{-1},W,Q hermiteska med lokalt integrerbara matriselement; och under vissa ytterligare villkor på P^{-1} och W. Under dessa förutsättningar härleds några huvudresultat beträf- fande asymptotiken, då z

ightarrowinfty, för Titchmarsh-Weyls M-funktion svarande mot nämnda ekvation. Vi kommer också att, i specialfallet av att P=W=I, formulera och bevisa några satser av Borg-Marchenko typ för den i detta fall aktuella ekvationen. Den andra ingående rapporten behandlar en högre ordningens ordinär differential ekvation på formen... (More)
Popular Abstract in Swedish

Denna avhandling utgöres av två separata rapporter, där studieobjektet för den första är en vektorvärd Sturm-Liouville ekvation på formen -(PU')'+QU=zU, xin[0,b), med P^{-1},W,Q hermiteska med lokalt integrerbara matriselement; och under vissa ytterligare villkor på P^{-1} och W. Under dessa förutsättningar härleds några huvudresultat beträf- fande asymptotiken, då z

ightarrowinfty, för Titchmarsh-Weyls M-funktion svarande mot nämnda ekvation. Vi kommer också att, i specialfallet av att P=W=I, formulera och bevisa några satser av Borg-Marchenko typ för den i detta fall aktuella ekvationen. Den andra ingående rapporten behandlar en högre ordningens ordinär differential ekvation på formen sum_{j,k=0}^{m}D^{j}a_{jk}D^{k}u=zu, xin[0,b), där D=id/dx, och där koefficien- terna a{jk}, j,kin[0,m], med a_{mm}=1, satisfierar vissa regularitetsvillkor och väljes så att matrisen (a_{jk}) blir hermitesk. Vi antar också att m>1. Medelst bruk av Paley-Wiener metoder kommer vi då att bevisa att motsvarande spektralmått bestämmer ekvationen entydigt upp till konjugering med en funktion av absolutbelopp 1. Vi kommer också att diskutera under vilka tilläggsvillkor som spektralmåttet entydigt bestämmer koefficenterna a_{jk}, j,kin[0,m], j+k<2m, såväl som b och randvillkoren i 0 och även ev. randvillkor i b. (Less)
Abstract
This discourse is constituted by two separate reprots, where the first one offers an elementary deduction of the leading order term asymptotics for the Titchmarsh-Weyl M-function corresponding to a vector-valued Sturm-Liouville equation of the form -(PU')'+QU=zu, xin[0,b), with P^{-1},W,Q being hermitean with locally integrable entries; and under some additional conditions on P^{-1} and W. In the special case of P=W=I, we give some further asymptotic results for the same M-function. In this case, we also prove that the corresponding spectral measure determines the equation uniquely up to conjugation by a constant and unitary matrix R, and we finish this presentation by giving a local form of the Borg-Marchenko theorem in the above case... (More)
This discourse is constituted by two separate reprots, where the first one offers an elementary deduction of the leading order term asymptotics for the Titchmarsh-Weyl M-function corresponding to a vector-valued Sturm-Liouville equation of the form -(PU')'+QU=zu, xin[0,b), with P^{-1},W,Q being hermitean with locally integrable entries; and under some additional conditions on P^{-1} and W. In the special case of P=W=I, we give some further asymptotic results for the same M-function. In this case, we also prove that the corresponding spectral measure determines the equation uniquely up to conjugation by a constant and unitary matrix R, and we finish this presentation by giving a local form of the Borg-Marchenko theorem in the above case (cf. [GS2, Chapter 3.]); a theorem which is due to Simon, [S], in the scalar case. The object of the second report is to study a higher order ordinary differential equation of the form sum_{j,k=0}^{m}D^{j}a_{jk}D^{k}=zu, xin[0,b), where D=id/dx, and where the coefficients a_{jk}, j,kin[0,m], with a_{mm}=1, satisfy certain regularity conditions and are chosen so that the matrix (a_{jk}) is hermitean. We will also assume that m>1. More precisely, we will prove, using Paley-Wiener methods, that the corresponding spectral measure determines the equation up to conjugation by a function of modulus 1. We will also discuss under which additional conditions the spectral measure uniquely determines the coefficients a_{jk}, j,kin[0,m], j+k<2m, as well as b and the boundary conditions at 0 and (if any) at b. (Less)
Please use this url to cite or link to this publication:
author
opponent
  • Prof W. Desmond Evans, W. Desmond Evans, Univ. of Wales, Cardiff, UK
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Functions, differential equations, Funktioner, differentialekvationer, Paley-Wiener., The generalized Fourier transform, A certain higher order ordinary differential equation, Borg-Marchenko theorems, Vector-valued Sturm-Liouville equations, Inverse spectral theory, Titchmarsh-Weyl M-function asymptotics, Asymptotics of solutions, Spectral measure
pages
3 pages
publisher
Center for Mathematical Sciences, Mathematics, Lund University, Box 118, SE-221 00 LUND, SWEDEN,
defense location
MH:C
defense date
2002-10-31 10:15
ISSN
1404-0034
ISBN
91-628-5313-9
language
English
LU publication?
yes
id
f386a4bc-987a-4ba9-bdff-61d667c76e6f (old id 465091)
date added to LUP
2007-09-27 14:38:18
date last changed
2016-09-19 08:44:59
@phdthesis{f386a4bc-987a-4ba9-bdff-61d667c76e6f,
  abstract     = {This discourse is constituted by two separate reprots, where the first one offers an elementary deduction of the leading order term asymptotics for the Titchmarsh-Weyl M-function corresponding to a vector-valued Sturm-Liouville equation of the form -(PU')'+QU=zu, xin[0,b), with P^{-1},W,Q being hermitean with locally integrable entries; and under some additional conditions on P^{-1} and W. In the special case of P=W=I, we give some further asymptotic results for the same M-function. In this case, we also prove that the corresponding spectral measure determines the equation uniquely up to conjugation by a constant and unitary matrix R, and we finish this presentation by giving a local form of the Borg-Marchenko theorem in the above case (cf. [GS2, Chapter 3.]); a theorem which is due to Simon, [S], in the scalar case. The object of the second report is to study a higher order ordinary differential equation of the form sum_{j,k=0}^{m}D^{j}a_{jk}D^{k}=zu, xin[0,b), where D=id/dx, and where the coefficients a_{jk}, j,kin[0,m], with a_{mm}=1, satisfy certain regularity conditions and are chosen so that the matrix (a_{jk}) is hermitean. We will also assume that m&gt;1. More precisely, we will prove, using Paley-Wiener methods, that the corresponding spectral measure determines the equation up to conjugation by a function of modulus 1. We will also discuss under which additional conditions the spectral measure uniquely determines the coefficients a_{jk}, j,kin[0,m], j+k&lt;2m, as well as b and the boundary conditions at 0 and (if any) at b.},
  author       = {Andersson, Erik},
  isbn         = {91-628-5313-9},
  issn         = {1404-0034},
  keyword      = {Functions,differential equations,Funktioner,differentialekvationer,Paley-Wiener.,The generalized Fourier transform,A certain higher order ordinary differential equation,Borg-Marchenko theorems,Vector-valued Sturm-Liouville equations,Inverse spectral theory,Titchmarsh-Weyl M-function asymptotics,Asymptotics of solutions,Spectral measure},
  language     = {eng},
  pages        = {3},
  publisher    = {Center for Mathematical Sciences, Mathematics, Lund University, Box 118, SE-221 00 LUND, SWEDEN,},
  school       = {Lund University},
  title        = {Titchmarsh-Weyl M-function asymptotics and some results in the inverse spectral theory for vector-valued Sturm-Liouville equations and a certain higher order ordinary differential equation},
  year         = {2002},
}