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There are many examples of potentials $V(x)$ for which Schrodinger's equation for a single particle in one dimension is exactly solvable, in the sense that we can give "nice" expressions for the eigenfunctions and eigenvalues of the operator $-\frac{1}{2}\partial_x^2+V(x)$; I do not know if there is a technical definition of "nice" here, but standard examples include the harmonic oscillator $V(x)=\frac{1}{2}x^2$, as well as the hydrogen atom, $V(x)=-1/x$. In some cases, there are also similarly nice expressions for time evolution under this operator.

Suppose we instead consider the operator $\partial_x D(x) \partial_x$, either on $\mathbb{R}$ or on an interval with some given boundary conditions. Are there known examples of functions $D(x)$, other than the trivial constant example, for which there are similarly nice expressions?

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  • $\begingroup$ You may want to exclude equations that can be transformed into the classical heat equation by means of strictly monotonic, smooth transformations - this is a powerful tool and allows one to find explicit solutions of even certain nonlinear problems, like Burger's equation, but is probably a bit trivial for your purposes. $\endgroup$
    – Delio Mugnolo
    Commented Feb 15, 2016 at 7:59

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