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Consider a Hamiltonian of the form

$$H=\frac{p_x^2}{2m}+\frac{p_y^2}{2M}+\frac{k(y)x^2}{2}$$

where ##k(y)## is some free function which I can choose at will. For ##k(y)=k=constant##, the Hamiltonian is a trivial combination of a free particle with position ##y## and a harmonic oscillator with position ##x##. However, I need the exact solution (of the Schrodinger equation) for some nontrivial case, when ##k(y)## is not a constant. Does anybody know such a function ##k(y)## for which the system can be solved exactly? A reference would be very welcome.

(I need it for a research paper on which I am working, so coautorship is also possible, in which case one can send me a private message. The research is related to the Casimir effect.)

$$H=\frac{p_x^2}{2m}+\frac{p_y^2}{2M}+\frac{k(y)x^2}{2}$$

where ##k(y)## is some free function which I can choose at will. For ##k(y)=k=constant##, the Hamiltonian is a trivial combination of a free particle with position ##y## and a harmonic oscillator with position ##x##. However, I need the exact solution (of the Schrodinger equation) for some nontrivial case, when ##k(y)## is not a constant. Does anybody know such a function ##k(y)## for which the system can be solved exactly? A reference would be very welcome.

(I need it for a research paper on which I am working, so coautorship is also possible, in which case one can send me a private message. The research is related to the Casimir effect.)

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