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¨
• dinger equations (reduced units)

−ψ′′(k, r) + V (r)ψ(k, r) = k2ψ(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−1/2F(−k)F −1(k)k1/2

• 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 12C + 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 (12C + α → 12C∗ + α) * reactions? (12C + α → 15N + 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¨
• dinger equation factorized as
• −d2/dr2 + V (r)
• ψ(k, r)

= k2ψ(k, r)

• A+A− − κ2

ψ(k, r) = k2ψ(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+ + κ2

A−ψ(k, r) = k2A−ψ(k, r)

• −d2/dr2 + ˜

V (r)

• ˜

ψ(k, r) = k2 ˜ ψ(k, r) * new potential ˜ V (r) = V (r) − 2U ′(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
• “Conservative” (usual) transformations: ˜

ϕ(k, r) = A−ϕ(k, r) * factorization solution at the origin σ(0) = 0 or ∞ * simple Jost-matrix transformation ˜ F(k) = (±κ − ık)±1F(k)

• iteration ⇒ arbitrary-order Pad´

e approximant, solves inverse problem

• problem: F diagonal ⇒ ˜

F diagonal (no coupling) ⇒ for single channels only!

• “Non-conservative” (new) transformations: ˜

ϕ(k, r) = A−ϕ(k, r) * 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

σ(r) = cosh(κr) + sinh(κr)κ−1U(0) = exp(κr)C + exp(−κr)D

• Transformed potential matrix ˜

V (r) = −2U ′(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 =   κ1 κ2   , U(0) =   α1 β β α2  

• Simplification: only three arbitrary parameters when

det C = det D = 0 ⇐ ⇒ α1,2 = ±

• (κ1κ2 − β2)(κ1/κ2)±1
• In this case, the determinant of the Jost matrix

˜ F(k1, k2) =  

k1+ıα1 k1−ıκ1 ıβ k2+ıκ2 ıβ k1−ıκ1 k2+ıα2 k2+ıκ2

  has zeros in k1 = ±

• κ2/κ1β − ı
• κ1/κ2
• κ1κ2 − β2 ⇒ possible resonance
• In terms of threshold ∆ = κ2

2 − κ2 1, resonance energy ER and width Γ

2κ2

1,2

=

• E2

R + Γ2/4 +

• (ER − ∆)2 + Γ2/4 ∓ ∆

4β4 =

• ER +
• E2

R + Γ2/4

ER − ∆ +

• (ER − ∆)2 + Γ2/4

Numerical example: ∆ = 10, ER = 7 and Γ = 1

V11 V12 V22 + ∆ Potential matrix r 2 1.5 1 0.5 10 5

• 5
• 10

δ2 ǫ δ1 Scattering matrix E 30 20 10 π π/2

V =   V11 V12 V12 V22   S =   cos sin − sin cos     exp(2ıδ1) exp(2ıδ2)     cos − sin 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)