Criterion of existence of eigenfunctions of spectral Darboux-Protter problems for a multidimensional Euler-Darboux-Poisson equation (Q1022258)

Let \(D_\varepsilon\) be a finite domain of the Euclidian space \(E_{m+1}\) of points \((x_1,x_2,\dots,x_m,t)\) bounded by \(|x|=t+\varepsilon\), \(|x|= 1-t\) and \(t=0\), where \(|x|=\sqrt{x_1^2+\cdots +x_m^2}\), \(0\leq t \leq (1-\varepsilon)/2\), \(0\leq \varepsilon< 1\). The aim of the paper is the spectral problem for the equation \[ L_\alpha u\equiv\Delta_xu- u_{tt}- \frac{\alpha}{t}u_t= \mu u \quad\text{in } D_\varepsilon, \] where \(\alpha \) and \(\mu \) are real parameters, with the edge conditions Problem 1: \[ \begin{aligned} u_\alpha |_S = 0,\;u_\alpha |_{S_\varepsilon}= 0 &\quad\text{for } \alpha< 1; \\ \frac{u_\alpha}{\ln t}|_S=0,\;u_\alpha|_{S_\varepsilon}= 0 &\quad\text{for } \alpha=1; \\ t^{\alpha-1}u_\alpha|_S=0,\;u_\alpha|_{S_\varepsilon}=0 &\quad\text{for }\alpha<1, \end{aligned} \] with \(\partial D_\varepsilon=S_\varepsilon\cup S_1\cup S\). Problem 2: \[ \begin{aligned} \frac{\partial u_\alpha}{\partial t}|_{t=0}=0,\;u_\alpha |_{S_\varepsilon}=0 &\quad\text{for }\alpha \geq 0 \\ \lim_{t \to 0}t^\alpha(u_\alpha- u_{\alpha,1})_t=0,\;u_\alpha |_{S_\varepsilon}=0 &\quad\text{for }\alpha<0; \end{aligned} \] where \(u_{\alpha,1}(x,t)\) is a solution of the Cauchy problem for the equation (1). The main result is the following: Problems 1 and 2 have for any \(\mu\) a countable set of eigenfunctions in the class \(C(\overline{D_\varepsilon}\setminus S) \cap C^2(D_\varepsilon)\) if and only if \(\varepsilon = 0\). The essential part of the proof is the reduction of the problem to a two-dimensional spectral Darboux problem.