How to solve this 2nd-order linear ODE analytically?


I want to analytically solve the eigenvalue problem $ $ y”(x) – 2\gamma\, y'(x) + [\lambda^2 + \gamma^2 – (\frac{x^2}{2}+\alpha)^2 + x]\, y(x)=0$ $ where $ \lambda$ is the eigenvalue and $ \alpha,\gamma$ are parameters. The boundary condition is $ y(\pm\infty)=0$ .

Or instead of the eigenvalue problem, it will as well be nice to just solve it with freely running $ \lambda$ . Then probably I can tackle the eigenproblem by imposing the boundary condition.

The following code doesn’t work well. Is there any possible way beyond?

F := (D[#, {x, 2}] -       2 \[Gamma] D[#, x] + (\[Lambda]^2 + \[Gamma]^2 + (x^2/2 + \[Alpha])^2 + x) #) &; DEigensystem[{F[y[x]] /. \[Lambda] -> 0,    DirichletCondition[y[x] == 0, True]},   y[x], {x, -\[Infinity], \[Infinity]}, 5] DSolve[F[y[x]] == 0, y[x], x]