Solving differential equation

Question

I have the following differential equation \begin{align} \left[-\partial^{2}_{x}+ x^{2} -1-2\alpha_{+}\right] \left[-\partial^{2}_{x}+ x^{2} -1-2\alpha_{-}\right] \phi(x)=0, \quad \alpha_{\pm}=\dfrac{-1}{2}+\dfrac{E^{2}}{2}\pm \dfrac{\sqrt{a^{4}+(\beta E)^{2}}}{2a^{2}} \end{align} My question is : what is the method used to find the eigenvalues of this equation. Thank you in advance.

Answer 1

This looks too much like a homework problem to warrant an answer, so I will remind you a couple of facts about the quantum harmonic oscillator and its solutions, the Hermite functions used in physics problems; your posting this here assumes you are intrigued by physics applications. Your second specification appears meaningless without further information.

You are interested for all all eigenfunctions $\phi(x)$ in the kernel of this operator, for allowable constants $\alpha_{\pm}$, $$ \hat O= \left[-\partial^{2}_{x}+ x^{2} -1-2\alpha_{+}\right] \left[-\partial^{2}_{x}+ x^{2} -1-2\alpha_{-}\right]. $$ As a physicist, you surely recognize in each square bracket Hermite's operator $$ D= (-\partial^{2}_{x}+ x^{2} -1)/2~, $$ trivially related to the oscillator hamiltonian. You know its eigenvalues are the integers, 0,1,2,...,n,... for eigenfunctions $\psi_n(x)$, the Hermite functions, linked, $D\psi_n = n\psi_n$, but I will not insist on integer ns in the absence of further (boundary) information.

Your operator then presents as $$ \hat O/4=(D-\alpha_+) (D-\alpha_-), $$ and, acting on these Hermite functions, $\psi_n(x) = (-1)^n \left (2^n n! \sqrt{\pi} \right)^{- 1/2} e^{ {x^2}/{2}} \partial_x^n e^{-x^2}$, it yields $$ (n-\alpha_+)(n-\alpha_-), $$ which must vanish for your kernel, so $n=\alpha_{\pm}$ will do.

You have to fuss the details and choices yourself, subject to the physical constraints of your problem, possibly continuing off the integers.