On one completeness condition for the root vectors of the Lavrent\('\)ev-Bitsadze equation (Q2713903)

Let \(\Omega\subset\mathbb R^2\) be a~bounded domain whose boundary consists of the Lyapunov curve~\(\sigma\) for \(y>0\) and the characteristics \(AC:x+y=0\) and \(BC:x-y=0\) for \(y<0\) of the operator NEWLINE\[NEWLINE Lu\equiv -\text{sgn }y\cdot u_{xx} -u_{yy}. NEWLINE\]NEWLINE The author considers in \(\Omega\) the Tricomi problem with spectral parameter NEWLINE\[NEWLINE (1) \quad Lu=\alpha u, \qquad (2) \quad u|_{\sigma\cup AC}=0.NEWLINE\]NEWLINE Denote by \(L_T\) the extension of \(L\) to the \(L_2(\Omega)\) space from the subspace of functions \(u(x,y)\) of the class \(C^2(\overline\Omega)\) satisfying condition~(2).NEWLINENEWLINENEWLINEThe main result of the paper reads: The set of eigenfunctions and associated eigenfunctions \(u_n(x,y)\) of \(L_T\) (Tricomi problems) is complete in \(L_2(\Omega)\) if and only if the set of their traces \(u_n|_{BC}\) is complete in \(L_2(BC)\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].