Consider a Schrödinger equation: $$-\frac{\text{d}^2}{\text{d}x^2}f(x)+U(x)f(x)=Ef(x),$$
I need a $U(x)$ satisfying the following:
- The Schrödinger equation with it must be solvable purely analytically, without need for any numerics (but using special functions, integrals or series is acceptable — the main point is that they must be explicit, not yet another equation to solve)
- $\displaystyle \lim_{x\to\infty} U(\pm x)=0$
- $\exists a,b: U(x)<0\;\forall x\in[a,b]$
I.e. $U(x)$ should represent some potential well, which would have both free and bound states.
Boundary conditions are imposed at some points $q$, $r$. Changing locations of these points shouldn't affect analytical solvability of the BVP.
Are there any such $U(x)$? If yes, what are examples?
Examples of what does not answer the question are:
- finite square potential well, because to solve it one has to solve transcendental equations, which need numerics
- $\delta$-shaped potential well, since despite it can be solved analytically for infinite space, it still results in transcendental equation when $x\in[q,r]$