Define the Hamiltonian $$ H = - \Delta + y^2 + a e^{b(x+y)}\,, $$ where $- \Delta = - \partial_x^2 - \partial_y^2$ and $a,b > 0$. I'm trying to determine the spectrum and/or generalized eigenfunctions of this operator. I guess there is some scattering theoretic methods to achieve this but I'm not very familiar with the literature. Especially the problem seems to be that the potential $V(x,y) = y^2 + a e^{b(x+y)}$ does not decay, which is often assumed in scattering theory books.
Here is some of my work.
First consider the case where we forget the exponential term: $$ H_0 = - \Delta + y^2\,. $$ The eigenvalue equation $H_0 f = E f$ is separable and we get the eigenfunctions $$ f_{p,n}(x,y) = g_p(x)h_n(y)\,, \quad p \in \mathbb{R}\,, \quad n \in \mathbb{N}\,. $$ where $g_p(x) = C_1 e^{ipx} + C_2 e^{-ipx}$ and $h_n$ are the eigenfunctions of the quantum harmonic oscillator Hamiltonian. The function $f_{p,n}$ corresponds to the eigenvalue $E(p,n) = p^2+n$ which leads us to believe that the spectrum of $H_0$ is $[0,\infty)$ and since $f_{p,n}$ are not in $L^2(\mathbb{R}^2)$ the spectrum should be fully continuous.
When we take into account the exponential term we expect that the solution of $Hf = Ef$ has the asymptotics \begin{align} f_{p,n}(x,y) &\to 0 \text{ as } x \to \infty\,,\\ f_{p,n}(x,y) & \sim \left( e^{ipx} + R(y,p) e^{-ipx} \right) h_n(y) \text{ as } x \to - \infty \end{align} where $R(y,p)$ is the reflection coefficient of the 1-dimensional scattering problem with potential $x \mapsto ae^{by}e^{bx}$ and now $p \geq 0$.
This leads me to believe that the spectrum of $H$ is $[0,\infty)$, fully continuous and I can "label" the (generalized) eigenfunctions with $\mathbb{R}_+ \times \mathbb{N}$. I'd like to know if this is true, and if it is, how to do this more rigorously.