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Let $\psi_\lambda(x, c)$, $\phi_\mu(y, {\tilde c})$ be the eigenfunctions of $H_x(c )$ and $H_y({\tilde c})$ respectively, $$ H_x \psi_\lambda(x, c) = \lambda(c) \psi_\lambda(x, c) $$ $$ H_y \phi_\mu(y, {\tilde c}) = \mu({\tilde c}) \phi_\mu(y, {\tilde c}) $$ Since both $\lbrace \psi_\lambda\rbrace_\lambda$ and $\{\phi_\mu\}_\mu$ must be basis sets, there ...



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