ON THE SPECTRAL PROPERTIES OF WAVE OPERATOR, PERTURBED BY THE LOWEST TERM

In this paper, we study the spectral properties of an operator beam for a wave equation perturbed by the lower-order term. The incorrectness of the minimal wave operator is well known, since zero is an infinite-to-one eigenvalue for it. In our work, we show that the situation changes if the operator is perturbed by a lower-order term containing the spectral parameter as a coefficient, and as a result, the studied problem takes the form of a bundle of operators. The resulting bundle of operators is easily factorized by first-order functional-differential operators which spectral properties are easily studied by the classical method of separation of variables. Direct application of the method of separation of variables to the original bundle of operators encounters the insurmountable difficulties. When adding a lower-order term with a spectral parameter and expanding the domain of definition, the task becomes solvable and the operator becomes reversible. In particular, it becomes reversible at certain values of the coefficient in the boundary condition.