An inverse problem is considered for an inhomogeneous Schr\"odinger equation.
Assuming that the potential vanishes outside a finite interval and satisfies
some other technical assumptions, one proves the uniqueness of the recovery of
this potential from the knowledge of the wave function at the ends of the above
interval for all energies. An algorithm is given for the recovery of the
potential from the above data

A. G. Ramm

Schrödinger Equation

English

arXiv

Ramm, Alexander G.

K-State Research Exchange

Inverse problem for an inhomogeneous Schrödinger equationA. G. Ramm  Citation: J. Math. Phys. 40, 3876 (1999); doi: 10.1063/1.532930 View online: http://dx.doi.org/10.1063/1.532930 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v40/i8 Published by the American Institute of Physics.  Related ArticlesSingularities for solutions to Schrödinger equations with asymptotically constant magnetic fields J. Math. Phys. 53, 073707 (2012) Quantum Painlevé-Calogero correspondence for Painlevé VI J. Math. Phys. 53, 073508 (2012) Quantum Painlevé-Calogero correspondence J. Math. Phys. 53, 073507 (2012) Riemann-Hilbert approach and N-soliton formula for coupled derivative Schrödinger equation J. Math. Phys. 53, 073506 (2012) On the complete integrability of a nonlinear oscillator from group theoretical perspective J. Math. Phys. 53, 073504 (2012)  Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors Downloaded 23 Jul 2012 to 129.130.37.175. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissionsJOURNAL OF MATHEMATICAL PHYSICS VOLUME 40, NUMBER 8 AUGUST 1999Inverse problem for an inhomogeneousSchro¨dinger equationA. G. Ramma)Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602~Received 8 March 1999; accepted for publication 27 April 1999!Let (l2k2)u52u91q(x)u2k2u5d(x), xPR, ]u/]uxu2iku0, uxu‘ . As-sume that the potential q(x) is real valued and compactly supported: q(x)5q(x), q(x)50 for uxu>1, *211 uqudx,‘ , and that q(x) produces no boundstates. Let u(21,k) and u(1,k), ;k.0 be the data. It is shown that under theabove assumptions these data determine q(x) uniquely. © 1999 American Insti-tute of Physics. @S0022-2488~99!02108-8#I. INTRODUCTIONFor several decades, the following inverse problems of practical interest are open. Let„2u1k2u1k2v~x !u52d~x !, in R3, ~1.1!u satisfies the radiation condition at infinity, and v(x) is a compactly supported piecewise-smoothfunction, supp v,R23 “$x:x3,0%.The data are the values u(x1 ,x2,0,k) for all xˆ“(x1 ,x2)PR2 and k.0.~IP1! The inverse problem is the following.Given the data, find v(x).Uniqueness of the solution to this problem is not proved. IP1 is not overdetermined: the datais a function of three variables, and v(x) is also.A similar inverse problem can be formulated: Let„2u1k2u2q~x !u50, in R3, ~1.2!u5eikax1A~a8,a ,k !eikrr1oS 1rD , r“uxu‘ , a85 xr, ~1.3!where aPS2 is a given unit vector, q(x) is a real-valued piecewise-smooth function,supp q(x),Ba“$x:uxu<a%, and S2 is the unit sphere.~IP2! Given A(a8,a0 ,k) for all a8PS2, all k.0 and a fixed a5a0PS2, find q(x).The uniqueness of the solution to ~IP2! is not proved.The third problem is the following.Let„2u1k2u2q~x !u52d~x !, in R3; ~1.4!u satisfies the radiation condition, and q(x) is the same as in ~IP2!.The data are the values u(x ,k)u uxu5a .~IP3! Given the data u(x ,k)u uxu5a for all k.0 and all x on the sphere Sa“$x:uxu5a%, findq(x).Uniqueness of the solution to ~IP3! is not proved.An overview of inverse problems and references one can find in Refs. 1–3.a!Electronic mail: ramm@math.ksu.edu38760022-2488/99/40(8)/3876/5/$15.00 © 1999 American Institute of PhysicsDownloaded 23 Jul 2012 to 129.130.37.175. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions3877J. Math. Phys., Vol. 40, No. 8, August 1999 Inverse problem for an inhomogeneous . . .Our purpose in this paper is to study the one-dimensional analog of ~IP3! and to prove for thisanalog a uniqueness theorem. The one-dimensional analog of ~IP3! corresponds to a plasmaequation in a layer.Letlu2k2u“2u91q~x !u2k2u5d~x !, xPR1, ~1.5!]u]uxu2iku0, uxu‘ . ~1.6!Assume that q(x) is a real-valued function,q~x !50, for uxu.1, qPL‘@21,1# . ~1.7!Suppose that the data,$u~21,k !,u~1,k !%, ;k.0, ~1.8!are given.The inverse problem analogous to ~IP3! is the following.~IP! Given the data ~1.8!, find q(x).This problem, as well as ~IP1!–~IP3!, is of practical interest. One can think about finding theproperties of an inhomogeneous slab ~the governing equation is a plasma equation! from theboundary measurements of the field, generated by a point source inside the slab.In the literature there are many results concerning various inverse problems for the homoge-neous version of Eq. ~1.5!, but it seems that no results concerning ~IP! are known.Assume that the self-adjoint operator l52d2/dx21q(x) in L2(R) has no negative eigenval-ues @this is the case when q(x)>0, for example#. The operator l is the closure in L2(R) of thesymmetric operator l0 defined on C0‘(R1) by the formula l0u52u91q(x)u . Our result is thefollowing.Theorem 1: Under the above assumptions IP has, at most, one solution.II. PROOF OF THEOREM 1The solution to ~1.5!–~1.6! isu5H g~k !@ f ,g# f ~x ,k !, x.0,f ~k !@ f ,g# g~x ,k !, x,0.~2.1!Here f (x ,k) and g(x ,k) solve homogeneous version of Eq. ~1.5! and have the following asymp-totics:f ~x ,k !;eikx, x1‘ , g~x ,k !;e2ikx, x2‘ , ~2.2!f ~k !“ f ~0,k !, g~k !“g~0,k !, ~2.3!@ f ,g#“ f g82 f 8g522ika~k !, ~2.4!where the prime denotes differentiation with respect to the x variable, and a(k) is defined by theequationf ~x ,k !5b~k !g~x ,k !1a~k !g~x ,2k !. ~2.5!Downloaded 23 Jul 2012 to 129.130.37.175. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions3878 J. Math. Phys., Vol. 40, No. 8, August 1999 A. G. RammIt is known ~see, for example, Ref. 4! thatg~x ,k !52b~2k ! f ~x ,k !1a~k ! f ~x ,2k !, ~2.6!a~2k !5a~k !, b~2k !5b~k !, ua~k !u2511ub~k !u2, kPR, ~2.7!a~k !511OS 1k D , k‘ , kPC1 ; b~k !5OS 1k D , uku‘ , kPR, ~2.78!@ f ~x ,k !,g~x ,2k !#52ikb~k !, @ f ~x ,k !,g~x ,k !#522ika~k !, ~2.8!a(k) is analytic in C1 , b(k), in general, does not admit analytic continuation from R, but if q(x)is compactly supported, then a(k) and b(k) are analytic functions of kPC\0.The functionsA1~k !“g~k ! f ~1,k !22ika~k ! , A2~k !“f ~k !g~21,k !22ika~k ! , ~2.9!are the data; they are known for all k.0. Therefore one can assume the functionsh1~k !“g~k !a~k ! , h2~k !“f ~k !a~k ! , ~2.10!to be known for all k.0, becausef ~1,k !5eik, g~21,k !5eik, ~2.11!as follows from the assumption ~1.7! and from ~2.2!.From ~2.10!, ~2.6!, and ~2.5!, it follows thata~k !h1~k !52b~2k ! f ~k !1a~k ! f ~2k !52b~2k !h2~k !a~k !1h2~2k !a~2k !a~k !,~2.12!a~k !h2~k !5b~k !a~k !h1~k !1a~k !h1~2k !a~2k !. ~2.13!From ~2.12! and ~2.13!, it follows that2b~2k !h2~k !1h2~2k !a~2k !5h1~k !, ~2.14!b~k !h1~k !1a~2k !h1~2k !5h2~k !. ~2.15!Eliminating b(2k) from ~2.14! and ~2.15!, one getsa~k !h1~k !h2~k !1a~2k !h1~2k !h2~2k !5h1~k !h1~2k !1h2~2k !h2~k !, ~2.16!ora~k !5m~k !a~2k !1n~k !, kPR, ~2.17!wherem~k !“2 h1~2k !h2~2k !h1~k !h2~k ! , n~k !“h1~2k !h2~k !1h2~2k !h1~k !. ~2.18!Downloaded 23 Jul 2012 to 129.130.37.175. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions3879J. Math. Phys., Vol. 40, No. 8, August 1999 Inverse problem for an inhomogeneous . . .Problem ~2.17! is a Riemann problem ~see Ref. 5 for the theory of this problem! for the pair$a(k),a(2k)%, the function a(k) is analytic in C1“$k:kPC,Im k.0% and a(2k) is analytic inC2 . The functions a(k) and a(2k) tend to one as k tends to infinity in C1 and, respectively, inC2 ; see Eq. ~2.78!.The function a(k) has finitely many simple zeros at the points ik j , 1< j<J , k j.0, where2k j2 are the negative eigenvalues of the operator l defined by the differential expression lu52u91q(x)u in L2(R).The zeros ik j are the only zeros of a(k) in the upper half-plane k.Defineind a~k !“ 12pi E2‘‘d ln a~k !. ~2.19!One hasind a5J , ~2.20!where J is the number of negative eigenvalues of the operator l, and, using ~2.10!, ~2.20! and~2.18!, one getsind m~k !522@ ind h1~k !1ind h2~k !#522@ ind g~k !1ind f ~k !22J# . ~2.21!Since l has no negative eigenvalues, it follows that J50.In this case ind f (k)5ind g(k)50 ~see Lemma 1 below!, so ind m(k)50, and a(k) isuniquely recovered from the data as the solution of ~2.17!, which tends to one at infinity; see Eq.~2.78!. If a(k) is found, then b(k) is uniquely determined by Eq. ~2.15!, and so the reflectioncoefficient r(k)“b(k)/a(k) is found. The reflection coefficient determines a compactly supportedq(x) uniquely ~see Ref. 2!.To make this paper self-contained, let us outline a proof of the last claim using an argumentdifferent from the one given in Ref. 2.If q(x) is compactly supported, then the reflection coefficient r(k)“b(k)/a(k) is meromor-phic. Therefore, its values for all k.0 determine uniquely r(k) in the whole complex k plane asa meromorphic function. The poles of this function in the upper half-plane are the numbers ik j ,j51,2,...,J . They determine uniquely the numbers k j , 1< j<J , which are a part of the standardscattering data $r(k),k j ,s j,1< j<J%, where s j are the norming constants.Note that if a(ik j)50 then b(ik j)Þ0: otherwise Eq. ~2.5! would imply f (x ,ik j)[0, incontradiction to the first relation ~2.2!.If r(k) is meromorphic, then the norming constants can be calculated by the formula s j52i@b(ik j)/ a˙(ik j)#52i Resk5ik j r(k), where the dot denotes differentiation with respect to k,and Res denotes the residue. So, for compactly supported potential, the values of r(k) for all k.0 determine uniquely the standard scattering data, that is, the reflection coefficient, the boundstates 2k j2, and the norming constants s j , 1< j<J . These data determine the potential uniquely.Theorem 1 is proved. hLemma 1: If J50 then ind f 5ind g50.Proof: We prove ind f 50. The proof of the equation ind g50 is similar. Since ind f (k) equalsthe number of zeros of f (k) in C1 , we have to prove that f (k) does not vanish in C1 . If f (z)50, zPC1 , then z5ik , k.0, and 2k2 is an eigenvalue of the operator l in L2(0,‘), with theboundary condition u(0)50.From the variational principle one can find the negative eigenvalues of the operator l inL2(R1) with the Dirichlet condition at x50 as consecutive minima of the quadratic functional.The minimal eigenvalue is2k25infE0‘@u821q~x !u2#dx“k0 , uPH˚ 1~R1!, iuiL2~R1!51, ~2.22!Downloaded 23 Jul 2012 to 129.130.37.175. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions3880 J. Math. Phys., Vol. 40, No. 8, August 1999 A. G. Rammwhere H˚ 1(R1) is the Sobolev space of H1(R1) functions, satisfying the condition u(0)50.On the other hand, if J50, then0<infE2‘‘@u821q~x !u2#dx“k1 , uPH1~R!, iuiL2~R!51. ~2.23!Since any element u of H˚ 1(R1) can be considered as an element of H1(R) if one extends u to thewhole axis by setting u50 for x,0, it follows from the variational definitions ~2.22! and ~2.23!that k1<k0 . Therefore, if J50, then k1>0, and therefore k0>0. This means that operator l onL2(R1) with the Dirichlet condition at x50 has no negative eigenvalues. This means that f (k)does not have zeros in C1 , if J50. Thus J50 implies ind f (k)50.Lemma 1 is proved. hRemark 2: The above argument shows that, in general,ind f <J and ind g<J , ~2.24!so that ~2.21! impliesind m~k !>0. ~2.25!Therefore the Riemann problem ~2.17! is always solvable.1 A. G. Ramm, Multidimensional Inverse Scattering Problems ~Longman/Wiley, New York, 1992!, pp. 1–385; Russiantranslation of the expanded monograph ~Mir, Moscow, 1994!, pp. 1–496.2 A. G. Ramm, ‘‘Property C for ODE and applications to inverse scattering,’’ Z. Anal. Angew. 18, 331–348 ~1999!.3 A. G. Ramm, Property C for ODE and Applications to Inverse Problems, to appear in Proceedings of the InternationalConference on Operator Theory, Fields Institute Communications.4 V. A. Marchenko, Sturm–Liouville Operators and Applications ~Birkha¨user, Boston, 1986!.5 F. Gakhov, Boundary Value Problems ~Pergamon, New York, 1966!.Downloaded 23 Jul 2012 to 129.130.37.175. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions

Inverse problem for an inhomogeneous
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Inverse problem for an inhomogeneous Schrödinger equation

Inverse problem for an inhomogeneous Schr\"odinger equation

Inverse problem for an inhomogeneous Schr\"odinger equation

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