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?



Exactly solvable examples of diffusion equation with variable diffusivity?
This would be the Schrödinger equation with a position dependent mass. Some exactly solvable examples are presented in 


* [Analytic results in the position-dependent mass Schrödinger problem](http://arxiv.org/abs/1306.0933)
* [Explicit solutions for N-dimensional Schrödinger equations with position-dependent mass](http://arxiv.org/abs/quant-ph/0512035)
* [Nonperturbative solutions for one-dimensional Schrödinger equation with position-dependent mass](http://onlinelibrary.wiley.com/doi/10.1002/qua.20861/pdf)
* [Series solutions of the Schrödinger equation with position-dependent mass for the Morse potential](http://www.sciencedirect.com/science/article/pii/S0375960104001203)


