On the resolvent existence and the separability of a hyperbolic operator with fast growing coefficients in $L_{2}(R^2)$

Abstract

This paper studies the question of the resolvent existence, as well as, the smoothness of elements from the definition domain (separability) of a class of hyperbolic differential operators defined in an unbounded domain with greatly increasing coefficients after a closure in the space $L_{2}(R^2)$. Such a problem was previously put forward by I.M. Gelfand for elliptic operators. Here, we note that a detailed analysis shows that when studying the spectral properties of differential operators specified in an unbounded domain, the behavior of the coefficients at infinity plays an important role.