Eigenfunction condition of time-independent Schrödinger equation

Question

I'm trying to get the eigenfunctions and eigenvalues of the following one-dimensional Schrodinger equation: $$ -\frac{1}{2}\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x)=E\psi(x)\,,$$ where the external potential reads:$$V(x)=2\tanh^2(x/\sqrt{2})\,.$$ Mathematica gives me the following analytical result for the eigenfunctions (I used the method sketched by @Stefen Luttrell in this post):

C[1] LegendreP[1/2 (-1 + Sqrt[17]), Sqrt[2] Sqrt[2 - e], Tanh[x]] + 
C[2] LegendreQ[1/2 (-1 + Sqrt[17]), Sqrt[2] Sqrt[2 - e], Tanh[x]]

and now the problem is to find the correct values for the coefficients $C[1]$ and $C[2]$. I want the wavefunction to vanish for $x\rightarrow \pm \infty$. I tried to use Asymptotic and Series functions, but in all cases Mathematica takes too much time to compute or I cannot get a clue for the coefficients behavior. Any help is greatly appreciated.

Answer 1

As pointed out by ydd, instead of looking at $x\to\pm\infty$ it is easier to change variables $y=\tanh(x)$ and look at $y\to\pm1$. So we want to check for which values of $\mu,\nu$ is the expression $c_1 L^\nu_\mu(y)+c_2 Q^\nu_\mu(y)$ finite at $y\to\pm1$, where $\mu=1/2 (-1 + \sqrt{17}), \nu=\sqrt{2(2-E)}$. I'm going to assume that $E<2$ so that $\nu$ is real, I leave it to you to explore other cases.

I use the following code to solve this. I write a and b instead of $\mu,\nu$ to simplify the notation.

First, the divergence at $y\to-1$,

Assuming[y > -1, Series[c1 LegendreP[a, b, y] + c2 LegendreQ[a, b, y], {y, -1, 0}] // Normal // FullSimplify]

I isolate the divergent term and obtain

c1 -> -(1/2) c2 π Cot[a π]

Next, the divergence at $y\to+1$,

Assuming[y < 1, Series[c1 LegendreP[a, b, y] + c2 LegendreQ[a, b, y] /. c1 -> -(1/2) c2 π Cot[a π], {y, 1, 0}] // Normal //   FullSimplify]

Finally, I isolate the divergence and get the condition

Csc[a π] Gamma[b] Sin[(a - b) π] == 0

If you write this in terms of $\mu,\nu$ you get a simple equation for $E$, essentially something like $E_n=1/4 (-1 + \sqrt{17} - 4 n + 4 \sqrt{17} n - 8 n^2)$ where $n$ is an arbitrary integer.

I most likely missed some solutions when performing algebraic manipulations (I was not careful with inverse functions and branch cuts) but you can double check that yourself.