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Group theoretical approach to the intertwined Hamiltonians. (English) Zbl 1024.81009
Let \(A= d/dx+ W(x)\), \(H= -d^2/dx^2+ V(x)\), and \(\widetilde H= -d^2/dx^2+\widetilde V(x)\). When \(AH= \widetilde HA\) holds, \(H\) and \(\widetilde H\) are intertwined Hamiltonians, having equivalent conditions: Riccati equations \(V= W^2- W'+\varepsilon\), \(\widetilde V= W^2+ W'+\varepsilon\). \(W= \phi'/\phi\) is derived from \(-\phi''+ (V(x)- \varepsilon)\phi= 0\). The authors give the generalized finite difference Bäcklund algorithm \[ \overline w(x)=- v(x)- 1/\{\gamma(x)^2(w(x)- v(x))\}+ \gamma'(x)/\gamma(x), \] by using the transformation group on the set of Riccati equations.
Their main theorem: Let \(\phi_w(x)\) be a solution of the equation \(-\phi''_w+ (V(x)- \varepsilon)\phi_w= 0\), \(0\leq\gamma(x)\in c^2(\text{Dom}(V(x)))\), and \(\phi_v(x)\) \((\neq\phi_w)\) a solution of the equation \(-\phi''_v+ (V(x)+ \gamma(x)^{-2}- \varepsilon)\phi_v= 0\). Then \(\phi(x)= \gamma(- d/dx+ v(x))\phi_w(x)\) with \(v(x)= \phi_v'/\phi_v\) satisfies the new equation \(-\phi''+ \{V(x)- 2((\gamma'/\gamma) v+v')+ \gamma''/\gamma- \varepsilon\}\phi= 0\).
Corollary: Suppose that \(-(\psi^{(0)}_n)''+ (V_0(x)- E_n) \psi^{(0)}_n= 0\) holds for \(n= 0,1,2,\dots\).
\(\phi^{(1)}_n(x)= (E_n- E_0)^{-1/2}(- d/dx+ w_1(x, E_0)) \psi^{(0)}_n\) with \(w_1(x, E_0)= (\psi^{(0)}_0)'/\psi^{(0)}_0\) for \(n= 1,2,\dots\) satisfy the equations \(-\phi''+ (V_1(x)- E_n)\phi= 0\) with \(V_1(x)= V_0(x)- 2w_1'(x, E_0)\) for \(n= 1,2,\dots\), respectively. Finally, they show oscillator-like potentials and Coulomb-like potentials as examples with non-constant \(\gamma(x)\).

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
39A99 Difference equations
81R99 Groups and algebras in quantum theory
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