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On the M-function and Borg-Marchenko theorems for vector-valued Sturm-Liouville equations

Andersson, Erik LU (2003) In Journal of Mathematical Physics 44(12). p.6077-6100
Abstract
We will consider a vector-valued Sturm-Liouville equation of the form R[U]:=-(PU')(')+QU=lambdaWU, xis an element of[0,b), with P-1, W, Qis an element ofL(loc)(1)([0,b))(mxm) being Hermitian and under some additional conditions on P-1 and W. We give an elementary deduction of the leading order term asymptotics for the Titchmarsh-Weyl M-function corresponding to this equation. In the special case of P=W=I, Qis an element ofL(1)([0,infinity))(mxm) and the Neumann boundary conditions at 0, we will also prove that M=(1/root-lambda) (I+R) (I-R)(-1), where R=lim(n-->infinity) R-n=Sigma(n=1)(infinity)Q(n), for recursively defined sequences {R-n} and {Q(n)}. If Qis an element ofL(loc)(1)([0,b))(mxm), 0<bless than or equal toinfinity, the... (More)
We will consider a vector-valued Sturm-Liouville equation of the form R[U]:=-(PU')(')+QU=lambdaWU, xis an element of[0,b), with P-1, W, Qis an element ofL(loc)(1)([0,b))(mxm) being Hermitian and under some additional conditions on P-1 and W. We give an elementary deduction of the leading order term asymptotics for the Titchmarsh-Weyl M-function corresponding to this equation. In the special case of P=W=I, Qis an element ofL(1)([0,infinity))(mxm) and the Neumann boundary conditions at 0, we will also prove that M=(1/root-lambda) (I+R) (I-R)(-1), where R=lim(n-->infinity) R-n=Sigma(n=1)(infinity)Q(n), for recursively defined sequences {R-n} and {Q(n)}. If Qis an element ofL(loc)(1)([0,b))(mxm), 0<bless than or equal toinfinity, the same formula is valid with an exponentially small error for large lambda. It is clear that expansions of this type are helpful in finding representatives of the KdV invariants. For P=W=I, we prove that the spectral measure corresponding to the equation R[U]=lambdaU uniquely determines Q as well as b and the boundary conditions at 0 and b. We finally give a new proof of a local form of the Borg-Marchenko theorem (cf. Gesztesy and Simon, "On local Borg-Marchenko uniqueness results," Commun. Math. Phys. 211, 273-287 (2000), Chap. 3); a theorem which is due to Simon [see Simon, "A new approach to inverse spectral theory, I. fundamental formalism," Ann. Math. 150, 1-29 (1999)] in the scalar case. For applications to physics, it is worth mentioning that vector-valued Sturm-Liouville equations appear in some problems in magneto-hydro-dynamics. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Mathematical Physics
volume
44
issue
12
pages
6077 - 6100
publisher
American Institute of Physics
external identifiers
  • wos:000186662300033
  • scopus:0344287499
ISSN
0022-2488
DOI
10.1063/1.1618922
language
English
LU publication?
yes
id
eb048a47-2222-42df-88ca-2810fa68f3e6 (old id 294971)
date added to LUP
2007-08-22 11:16:43
date last changed
2017-01-01 07:27:54
@article{eb048a47-2222-42df-88ca-2810fa68f3e6,
  abstract     = {We will consider a vector-valued Sturm-Liouville equation of the form R[U]:=-(PU')(')+QU=lambdaWU, xis an element of[0,b), with P-1, W, Qis an element ofL(loc)(1)([0,b))(mxm) being Hermitian and under some additional conditions on P-1 and W. We give an elementary deduction of the leading order term asymptotics for the Titchmarsh-Weyl M-function corresponding to this equation. In the special case of P=W=I, Qis an element ofL(1)([0,infinity))(mxm) and the Neumann boundary conditions at 0, we will also prove that M=(1/root-lambda) (I+R) (I-R)(-1), where R=lim(n--&gt;infinity) R-n=Sigma(n=1)(infinity)Q(n), for recursively defined sequences {R-n} and {Q(n)}. If Qis an element ofL(loc)(1)([0,b))(mxm), 0&lt;bless than or equal toinfinity, the same formula is valid with an exponentially small error for large lambda. It is clear that expansions of this type are helpful in finding representatives of the KdV invariants. For P=W=I, we prove that the spectral measure corresponding to the equation R[U]=lambdaU uniquely determines Q as well as b and the boundary conditions at 0 and b. We finally give a new proof of a local form of the Borg-Marchenko theorem (cf. Gesztesy and Simon, "On local Borg-Marchenko uniqueness results," Commun. Math. Phys. 211, 273-287 (2000), Chap. 3); a theorem which is due to Simon [see Simon, "A new approach to inverse spectral theory, I. fundamental formalism," Ann. Math. 150, 1-29 (1999)] in the scalar case. For applications to physics, it is worth mentioning that vector-valued Sturm-Liouville equations appear in some problems in magneto-hydro-dynamics.},
  author       = {Andersson, Erik},
  issn         = {0022-2488},
  language     = {eng},
  number       = {12},
  pages        = {6077--6100},
  publisher    = {American Institute of Physics},
  series       = {Journal of Mathematical Physics},
  title        = {On the M-function and Borg-Marchenko theorems for vector-valued Sturm-Liouville equations},
  url          = {http://dx.doi.org/10.1063/1.1618922},
  volume       = {44},
  year         = {2003},
}