Inverse Scattering Problem And Generalized Optical Theorem
Inverse Scattering problemandgeneralized optical theorem@ ICNT workshop, MSU, 28 May 2015Kazuo Takayanagi and Mariko Oishi, Sophia U, Japan
contents1.2.3.4.5.6.7.8.IntroductionCurrent theory of inverse scatteringGeneralized optical theoremTheory of inverse scatteringTest calculation in one-dimensiont(k) of separable potentialt(k) of Gaussian potentialSummary
1. Introduction--- history of inverse problemsJ.W.S.Rayleigh, 1877Given eigenfrequencies of a string,can one determine the density distribution of the string ?The theory of sound, Dover, 1945Marc Kac, 1966Can one hear the shape of a drum ?Am.Math.Monthly 73, No.4, part II (1966) 1Inverse scattering in 1D ( 1960’s)δ (k)one-to-oneV (r)Inverse scattering in 3D (2015, present work)’one-to-many
2. Current theory of inverse scattering (Marchenko)inputδ (k) : phase shiftsource termone-to-oneδ (k)V (r)function of a single variableMarchenko eq.potentialTextbook:R.G.Newton, Scattering Theory of Waves and Particles,2nd edition, Springer,1982K.Chadan and P.C.Sabatier,Inverse Problems in Quantum Scattering Theory,2nd edition, Springer,1989
exampleGaussian potentialCan this be the end of the story ?Phase shift δ (k)V (r)ScatteringThe answer isInversescatteringNO―exactMarchenko
2-2. Why NO ? --- current status vs. present theoryCurrent status Formulation in one-dimension via partial wave decompositionAssumption of local potential V (r)excludes nonlocal V (r,r’)Physical meaning is not clear(Theory in coordinate space)Present theory Direct formulation in three-dimensional spaceno assumption of symmetry General nonlocal potential V (r,r’) is included Physical meaning is clear (Theory in momentum space)
3. Generalized Optical Theorem （GOT)3-1. Scattering theory in momentum spaceT-matrix: T-operator: Half-on-shell (HOS) T-matrix: On-shell (OS) T-matrixScattering statek' kk' k
?3-2. How much is arbitrary inQuestionAnswerLS eq.To beT-matrixphysicallyacceptable,HOSis closerto observables than pot.t(k) opticalk ' T ktheoremk ' potentialV k HOSobservablesT-matrixk ' T k ?Hermitian?GeneralizedOptical Theorem (GOT)studied only forM.Baranger et.al., Nucl.Phys.A138(1961)1single equation only is known forIts meaning is, however, unknown.F.E.Low, Phys.Rev.97,1392(1955)K.Takayanagi, Phys.Rev.A77, 062714(2008)Low equation
3-3. Condition for HOS T-matrix (no bound state, for simplicity)Hermitian V (r,r’)Generally nonlocalCompleteness ofAorthogonality ofB Be careful.Product of singular factors.
3-4. Generalized optical theorem (GOT)Completeness and orthogonality of scattering statesAintegral over on-shell momentumLow equationBintegral over off-shell momentumA BOptical theoremare independent equationsGeneralized optical theorem (GOT)clear physical meaning-- orthonormality ofcoupled set of nonlinear integral eqs. forHOS T-matrix
4. Theory of Inverse scattering4-1. on-to-one correspondence between V and T: LS equationPrepare Hermitian potentialSolve LS equation forfrom V to THOS T-matrixpotentialHermitianfrom T to VGeneralizedoptical theorem A BPrepare HOS T-matrix that satisfies A BSolve LS equation for
scattering problemproblem in momentum space4-2. inverse scatteringextension of T from OS to HOS region, satisfying A5 variablesB6 variablesA BHOSOSLSA B do not uniquely determine k k k'A Bt(k) k'k' T kk' T ' kt(k)kk'' VV' kk
4-3. how to solve A (and B ) for a given t(k)A nonlinear eq. for fixiterative solutionvia: OS part fixed A becomes a linear eq. forOS part of obtained replace OS part of obtainedsolve forgiven t(k)with given t(k)for next iteration k k'k ' T init kt(k) kAk' T k k' kt(k) k'k' T kt(k)
4-4. solution to inverse scattering problem k k'k' T k kA k'BConstraint on potential Potential be separable Potential be local etct(k)k' T kt(k) k k'desiredk' T kdesiredk' V k
5. Test calculation in one-dimensionS-wave, for simplicity.t(k)k' V k1 variables2 variablesCalculation with/without constraint on t(k) of separable potential t(k) of gaussian potentialk' V k
6. t(k) of separable potential6-1. calculation without constraintSeparable potential to generate the target t(k)Calculation without constraint on the solution k' V kt(k)k' V kConvergent results depend on initial inputA Bt(k)
calculation without constraintpotentialcoincideL-S eq.InputPhase shiftAt(k)Half-on-Shell T-matrixBHalf-on-Shell T-matrix
calculation without constraintpotentialdifferentL-S eq.Inputp0 1Phase shiftAt(k)Half-on-Shell T-matrixBHalf-on-Shell T-matrix
calculation without constraintpotentialdifferentL-S eq.Inputp0 3Phase shiftAt(k)Half-on-Shell T-matrxBHalf-on-Shell T-matrix
calculation without constraintpotentialdifferentL-S eq.Inputp0 10Phase shiftAt(k)Half-on-Shell T-matrixBHalf-on-Shell T-matrix
Comparing potentialsseparablepotentialseparablePhase shiftinputanalytick’Ap0 1Bkp0 3p0 10
6-2. calculation with constraintSeparable potential to generate the target t(k)one-to-oneCalculation with constraintt(k)Potential be separableConvergent results do not depend on initial inputseparableA Bt(k)δ (k)
calculation with constraintpotentialcoincideL-S eq.Inputp0 1Phase shiftAt(k)Half-on-Shell T-matrixBHalf-on-Shell T-matrixseparable
calculation with constraintpotentialcoincideL-S eq.Inputp0 3Phase shiftAt(k)Half-on-Shell T-matrixBHalf-on-Shell T-matrixseparable
calculation with constraintpotentialcoincideL-S eq.Inputp0 10Phase shiftAt(k)Half-on-Shell T-matrixBHalf-on-Shell T-matrixseparable
Comparing potentialsseparablepotentialseparablePhase shiftinputk’BAseparablekp0 1p0 3p0 10
7. t(k) of Gaussian potentialGaussian potential to generate the target t(k)Calculation without constraintCalculation with constraintPotential be separablePhase shift of a Gaussian potentialbe reproduced by a separable potential
Comparing potentialsGaussianpotentialexactPhase shiftinputBAseparablep0 1p0 3p0 10
8. summaryGeneralized optical theorem (GOT) Condition for physically acceptable HOS T-matrix AB Clear physical meaningAcompletenessof scattering statesBorthogonality Physics starting from Tk' V kk' T kobservableTheory of inverse scattering in 3D space Direct solution in 3D space. No symmetry assumption is necessary. Maximum degrees of freedom to reproduce t(k)solution to A B with arbitrary constraintconstraintA Bt(k)Successful demonstrationdesireddesired
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Inverse Scattering problem and generalized optical theorem @ ICNT workshop, MSU, 28 May 2015 Kazuo Takayanagi and Mariko Oishi, Sophia U, Japan . contents 1. Introduction 2. Current theory of inverse scattering . Inverse Problems in Quantum Scattering Theory, 2nd edition, Springer,1989 . Gaussian potential
SCATTERING AND INVERSE SCATTERING ON THE LINE FOR A . via the so-called inverse scattering transform method. The direct and inverse problems for the corresponding first-order linear sys-tem with energy-dependent potentials are investigated. In the direct problem, when . In quantum mechanics, ei .
I The phaseless inverse scattering problem for the Schr odinger equation was posed in the book of K. Chadan and P.C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer-Verlag, New York, 1977 I It was also implicitly posed in the book of R.G. Newton, Inverse Schr odinger Scattering in Three Dimensions, Springer, New York, 1989
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