# Can this Helmholtz PDE with Robin boundary conditions be solved analytically?

Consider the following Helmholtz problem in the infinite triangle $$y>0,\;x>y$$ with parameters $$Q<0$$, $$P\ge0$$, $$P<|Q|$$.

\left\{\begin{align} &\psi^{(2,0)}(x,y)+\psi^{(0,2)}(x,y)+E\psi(x,y)=0,\\ &\psi^{(0,1)}(x,0)-\frac Q2\psi(x,0)=0,\\ &\psi^{(1,0)}(x,x)-\psi^{(0,1)}(x,x)-\frac P{\sqrt2}\psi(x,x)=0\\ &|\psi(x,y)|<\infty. \end{align}\right.\tag1

I'm mainly interested in the solution with lowest $$E$$. In the case of $$P=0$$ it's easy to see that

$$\psi_0(x,y)=\exp\left(\frac Q2 (x+y)\right)\tag2$$

with eigenvalue

$$E=-\frac{Q^2}2\tag3$$

solves the problem. But what about $$P>0$$? Can $$(1)$$ still be solved analytically (i.e. in terms of elementary or special functions)? If not, can the solution be given in the form of an integral or a series with explicitly specified terms?