Simplest Schrödinger equation with both continuous and residual spectrum

Simplest Schrödinger equation with both continuous and residual spectrum

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 is acceptable)
$\displaystyle \lim_{x\to\infty} U(\pm x)=0$
$\exists a,b: U(x)<0\;\forall x\in[a,b]$
I.e. $U(x)$ should represent some potential well, which would have both
free and bound states.
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 $x\in[q,r]$