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


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?


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