Supersymmetric Quantum Mechanics for Coupled-Channel Systems - PowerPoint PPT Presentation

Supersymmetric Quantum Mechanics for Coupled-Channel Systems Jean-Marc Sparenberg PNTPM, Universit´ e Libre de Bruxelles In collaboration with • Daniel Baye (PNTPM) • Boris Samsonov (Tomsk State University, Russia) • Fran¸ cois Foucart (Cornell University, USA)

Outline • Definitions: coupled-channel quantum scattering theory • Motivation: inverse scattering problem for coupled channels • Tool: supersymmetric-(quantum-mechanics-)transformations • Two types of supersymmetric transformations * usual transformations: no coupling * new transformations: coupling • Application: new transformation of vanishing potential ⇒ exactly-solvable coupled-channel model of Feshbach resonance No physical application (yet)!

Definitions: coupled-channel quantum scattering theory • Coupled radial Schr¨ odinger equations (reduced units) − ψ ′′ ( k, r ) + V ( r ) ψ ( k, r ) = k 2 ψ ( k, r ) (1) * V = potential matrix, ψ = general solution matrix/vector * complex wave number matrix k = diag √ E − ∆ i , center-of-mass energy E * threshold energies ∆ 1 = 0 ≤ ∆ 2 ≤ . . . ≤ ∆ N for N channels • Regular solution matrix ϕ : solution of Eq. (1) vanishing at the origin • Jost solution matrices f : solutions of (1) satisfying f ( ± k, r ) ∼ r →∞ exp( ± ıkr ) • Jost matrix F (key quantity of scattering theory) defined by ϕ ( k, r ) ∝ f ( − k, r ) F ( k ) + f ( k, r ) F ( − k ) * bound/resonant states = zeros of determinant in upper/lower half k plane * scattering matrix S ( k ) = k − 1 / 2 F ( − k ) F − 1 ( k ) k 1 / 2 • In general, V non diagonal (coupling) ⇐ ⇒ F non diagonal

Motivation: inverse scattering problem for coupled channels direct problem − → V ( r ) S ( k ) ← − inverse problem • Simplest inversion method: fit by phenomenological potential * example: Woods-Saxon potential for 12 C + α elastic scattering data * problem: high-quality fit not guaranteed but needed for low-energy extrapolation and error estimates (e.g. nuclear astrophysics) • Deductive methods: no assumption on potential, iteration until perfect fit * example: supersymmetric-quantum-mechanics method [ Inverse scattering with Supersymmetric Quantum Mechanics , Baye & Sparenberg, JPA 2004] * problem : only works for single channel (spin zero, elastic scattering) • However, numerous needs for coupled-channel models in nuclear physics * spin larger than zero (neutron + proton) * inelastic scattering ( 12 C + α → 12 C ∗ + α ) * reactions? ( 12 C + α → 15 N + p)

Supersymmetric- (Darboux-)transformation principle • Algebraic technique (based on superalgebra) allowing to transform an initial potential V into a new potential ˜ V • Initial radial Schr¨ odinger equation factorized as − d 2 /dr 2 + V ( r ) k 2 ψ ( k, r ) � � ψ ( k, r ) = A + A − − κ 2 � � k 2 ψ ( k, r ) ψ ( k, r ) = (2) A ± = ± d/dr + U ( r ), superpotential U ( r ) = σ ′ ( r ) σ − 1 ( r ) with σ = solution of Eq. (1) at negative factorization energy E (wave number κ = diag √ ∆ i − E ) • Transformed equation obtained by applying A − on the left to Eq. (2) A − A + + κ 2 � A − ψ ( k, r ) k 2 A − ψ ( k, r ) � = � − d 2 /dr 2 + ˜ � k 2 ˜ ˜ V ( r ) ψ ( k, r ) = ψ ( k, r ) * new potential ˜ V ( r ) = V ( r ) − 2 U ′ ( r ) * new solutions ˜ ψ ( k, r ) = A − ψ ( k, r ) = − ψ ′ ( k, r ) + U ( r ) ψ ( k, r )

Two types of supersymmetric transformations • Conserving or breaking boundary condition at the origin ϕ ( k, r ) = A − ϕ ( k, r ) • “Conservative” (usual) transformations: ˜ * factorization solution at the origin σ (0) = 0 or ∞ * simple Jost-matrix transformation ˜ F ( k ) = ( ± κ − ık ) ± 1 F ( k ) - iteration ⇒ arbitrary-order Pad´ e approximant, solves inverse problem - problem : F diagonal ⇒ ˜ F diagonal (no coupling) ⇒ for single channels only! ϕ ( k, r ) � = A − ϕ ( k, r ) • “Non-conservative” (new) transformations: ˜ * factorization solution at the origin σ (0) finite * complicated Jost-matrix transformation F ( k ) = ( ± κ − ık ) − 1 � � F ( k ) U (0) − f ′ T ( k, 0) ˜ - iterations? - F, f diagonal, U (0) non diagonal ⇒ ˜ F non diagonal (coupling)

Application: non-conservative transformation of V = 0 • Initial potential matrix V ( r ) = 0 • Regular solution matrix ϕ ( k, r ) ∝ sin( kr ) Jost solution matrices f ( ± k, r ) = exp( ± ıkr ) ⇒ Jost matrix F ( k ) = Id • Factorization solution matrix at negative factorization energy cosh( κr ) + sinh( κr ) κ − 1 U (0) σ ( r ) = = exp( κr ) C + exp( − κr ) D � ′ • Transformed potential matrix ˜ V ( r ) = − 2 U ′ ( r ) = − 2 � σ ′ ( r ) σ − 1 ( r ) * exactly solvable * depends on N ( N + 3) / 2 arbitrary parameters for N channels * equivalent to potential of [Cox, JMP 1964] for det C � = 0 * here, simpler writing and generalization: no condition on C

Non-conservative transformation of V = 0 (simplifying case) • For N = 2 channels, five arbitrary real parameters      κ 1 0  α 1 β  , κ = U (0) =  0 κ 2 β α 2 • Simplification: only three arbitrary parameters when � ( κ 1 κ 2 − β 2 )( κ 1 /κ 2 ) ± 1 det C = det D = 0 ⇐ ⇒ α 1 , 2 = ± • In this case, the determinant of the Jost matrix   k 1 + ıα 1 ıβ ˜ k 1 − ıκ 1 k 2 + ıκ 2 F ( k 1 , k 2 ) =   ıβ k 2 + ıα 2 k 1 − ıκ 1 k 2 + ıκ 2 κ 1 κ 2 − β 2 ⇒ possible resonance � � � has zeros in k 1 = ± κ 2 /κ 1 β − ı κ 1 /κ 2 • In terms of threshold ∆ = κ 2 2 − κ 2 1 , resonance energy E R and width Γ � ( E R − ∆) 2 + Γ 2 / 4 ∓ ∆ � 2 κ 2 E 2 R + Γ 2 / 4 + = 1 , 2 � � � � � ( E R − ∆) 2 + Γ 2 / 4 � 4 β 4 E 2 R + Γ 2 / 4 = E R + E R − ∆ +

Numerical example: ∆ = 10 , E R = 7 and Γ = 1 Scattering matrix Potential matrix π 10 V 22 + ∆ 5 δ 1 V 12 π/ 2 ǫ 0 V 11 -5 δ 2 0 -10 0 10 20 30 0 0.5 1 1.5 2 r E    V 11 V 12 V =  V 12 V 22       cos � sin �  exp(2 ıδ 1 ) 0  cos � − sin � S =     − sin � cos � 0 exp(2 ıδ 2 ) sin � cos � • Simple(st) exactly-solvable model for * Feshbach resonance * threshold cusp effect • Non-trivial coupling, as expected

Conclusions • New type of supersymmetric-quantum-mechanics transformation, breaking boundary conditions (“non-conservative”) ⇒ coupled potential models from uncoupled ones • Application to V = 0 ⇒ exactly-solvable ˜ V * simple model for Feshbach resonance, threshold cusp effect * promising first step for coupled-channel inverse problem • Interest for coupled-channel inversion both in nuclear (non-zero spins, inelastic scattering, reactions) and atomic physics (magnetic Feshbach resonance in atom-atom interactions)