On the Mfunction and BorgMarchenko theorems for vectorvalued SturmLiouville equations
(2003) In Journal of Mathematical Physics 44(12). p.60776100 Abstract
 We will consider a vectorvalued SturmLiouville equation of the form R[U]:=(PU')(')+QU=lambdaWU, xis an element of[0,b), with P1, W, Qis an element ofL(loc)(1)([0,b))(mxm) being Hermitian and under some additional conditions on P1 and W. We give an elementary deduction of the leading order term asymptotics for the TitchmarshWeyl Mfunction 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/rootlambda) (I+R) (IR)(1), where R=lim(n>infinity) Rn=Sigma(n=1)(infinity)Q(n), for recursively defined sequences {Rn} 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 vectorvalued SturmLiouville equation of the form R[U]:=(PU')(')+QU=lambdaWU, xis an element of[0,b), with P1, W, Qis an element ofL(loc)(1)([0,b))(mxm) being Hermitian and under some additional conditions on P1 and W. We give an elementary deduction of the leading order term asymptotics for the TitchmarshWeyl Mfunction 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/rootlambda) (I+R) (IR)(1), where R=lim(n>infinity) Rn=Sigma(n=1)(infinity)Q(n), for recursively defined sequences {Rn} 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 BorgMarchenko theorem (cf. Gesztesy and Simon, "On local BorgMarchenko uniqueness results," Commun. Math. Phys. 211, 273287 (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, 129 (1999)] in the scalar case. For applications to physics, it is worth mentioning that vectorvalued SturmLiouville equations appear in some problems in magnetohydrodynamics. (Less)
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http://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
 external identifiers

 wos:000186662300033
 scopus:0344287499
 ISSN
 00222488
 DOI
 10.1063/1.1618922
 language
 English
 LU publication?
 yes
 id
 eb048a47222242df88ca2810fa68f3e6 (old id 294971)
 date added to LUP
 20070822 11:16:43
 date last changed
 20180529 11:58:39
@article{eb048a47222242df88ca2810fa68f3e6, abstract = {We will consider a vectorvalued SturmLiouville equation of the form R[U]:=(PU')(')+QU=lambdaWU, xis an element of[0,b), with P1, W, Qis an element ofL(loc)(1)([0,b))(mxm) being Hermitian and under some additional conditions on P1 and W. We give an elementary deduction of the leading order term asymptotics for the TitchmarshWeyl Mfunction 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/rootlambda) (I+R) (IR)(1), where R=lim(n>infinity) Rn=Sigma(n=1)(infinity)Q(n), for recursively defined sequences {Rn} 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 BorgMarchenko theorem (cf. Gesztesy and Simon, "On local BorgMarchenko uniqueness results," Commun. Math. Phys. 211, 273287 (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, 129 (1999)] in the scalar case. For applications to physics, it is worth mentioning that vectorvalued SturmLiouville equations appear in some problems in magnetohydrodynamics.}, author = {Andersson, Erik}, issn = {00222488}, language = {eng}, number = {12}, pages = {60776100}, publisher = {American Institute of Physics}, series = {Journal of Mathematical Physics}, title = {On the Mfunction and BorgMarchenko theorems for vectorvalued SturmLiouville equations}, url = {http://dx.doi.org/10.1063/1.1618922}, volume = {44}, year = {2003}, }