A spectral perturbation problem and its applications to waves above an underwater ridge (Q2773670)

Under consideration is the following eigenvalue problem: NEWLINE\[NEWLINE (\lambda + T)u=P[qLu+q^2Mu], \tag{1}NEWLINE\]NEWLINE where \(T,P\) are one-dimensional pseudodifferential operators of finite order with symbols \(t(\xi)\) and \(p(\xi)\), \(q\) is a small parameter, and the linear integral operators \(L,M\) are of the form NEWLINE\[NEWLINE \widehat{Lu}(\xi)=l_1(\xi)\int l_1(\eta)l_2(\xi,\eta)\widehat{u}(\eta) d\eta, \quad \widehat{Mu}(\xi)=m_1(\xi)\int m_2(\xi,\eta)\widehat{Nu}(\eta) d\eta, NEWLINE\]NEWLINE with \(N\) a linear operator. Let \(\Pi_{\rho}=\{\xi\in {\mathbb C}:\;|\text{Im} \xi|\leq \rho\}\). All the symbols are assumed to be analytic functions of \(\eta,\xi\in \Pi_{\rho_0}\) for some \(\rho_0>0\). The symbols \(t(\xi)\), \(p(\xi)\), \(l_1(\xi)\), and \(l_2(\xi,\eta)\) are real for real \(\xi,\eta\) and, thus, the corresponding operators are selfadjoint. Moreover, \(\text{Im} t(\xi)=\text{Im} p(\xi)=0\) for \(\xi\in \Pi_{\rho_0}\) such that \(\text{Re} \xi=0\). The operator \(M\) is a subordinate operator, in particular, the equality \(Mu\equiv 0\) is possible. Under certain additional conditions, it is proven that the equation (1) has a solution for all sufficiently small parameters \(q\). This solution is sought in the space defined as a completion of \(C_0^{\infty}({\mathbb R})\) with respect to the norm NEWLINE\[NEWLINE \|\varphi\|_s^2=\sup\limits_{|\text{Im} \xi|<\rho} \int |1+\xi^2|^{s/2}|\widehat{\varphi}(\xi)|d\operatorname {Re}\xi. NEWLINE\]NEWLINE Note that the operator \(T\) has no discrete spectrum and the problem in question is actually the eigenvalue problem for the perturbation of \(T\). The results are applied to the problem of describing surface capillary-gravity waves of small amplitude in an ideal incompressible irrotational fluid.