On the M-function and Borg-Marchenko theorems for vector-valued Sturm-Liouville equations
(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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https://lup.lub.lu.se/record/294971
- author
- Andersson, Erik LU
- organization
- publishing date
- 2003
- 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 (AIP)
- 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
- 2016-04-01 17:13:52
- date last changed
- 2022-01-29 01:17:10
@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-->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.}}, author = {{Andersson, Erik}}, issn = {{0022-2488}}, language = {{eng}}, number = {{12}}, pages = {{6077--6100}}, publisher = {{American Institute of Physics (AIP)}}, 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}}, doi = {{10.1063/1.1618922}}, volume = {{44}}, year = {{2003}}, }