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 L2(R2). 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.