A differential equation for Lerch's transcendent and associated symmetric operators in Hilbert space

The function $\Psi(x, y, s)=e^{iy}\Phi(-e^{iy},s,x)$, where $\Phi(z,s,v)$ is Lerch's transcendent, satisfies the following two-dimensional formally self-adjoint second-order hyperbolic differential equation:
$$ L[\Psi]=\frac{\partial^2\Psi}{\partial x\,\partial y}+i(x-1)\frac{\partial\Psi}{\partial x}+\frac{i}{2}\Psi=\lambda\Psi, $$
where $s={1}/{2}+i\lambda$. The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space $L_2(\Pi)$, where $\Pi=(0,1)\times(0,2\pi)$. We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of $\Psi(x,y,s)$. We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function.
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