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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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