Spectral problem for a mathematical model of hydrodynamics

Spectral problems of the form $(T+P)u=\lambda u$ have a huge range of applications: hydrodynamic stability problems, elastic vibrations of a membrane, a set of possible states of systems in quantum mechanics, and so forth. The self-adjoint operators perturbed by bounded operators are most thoroughly studied. In applications, the perturbed operator is usually represented by the Sturm–Liouville or Schrodinger operator. At present moment, the researchers are very interested in the equations not solved with respect to the highest derivative $L\dot u=Tu+f$, which are known as Sobolev type equations. The study of Sobolev type equations leads to spectral problems of the form $T u = \lambda Lu$. In many cases, the operator $T$ can be perturbed by an operator $P$, and then the spectral problem takes the form $(T + P)u = \lambda Lu$. The study of such problems allows to construct a solution of the equation, as well as to investigate various parameters of mathematical models. Previously, such spectral problems with the perturbed operator were not studied. In this paper, we propose the method for investigating and solving the direct spectral problem for a hydrodynamic model.