On the bounded invertibility of a Schr¨odinger operator with a negative parameter in the space L2(Rn)

Abstract

The Schrodinger operator L = — Д + q(x), x £ Rn, is one of the main operators of modern quantum mechanics and theoretical physics. It is known that many fundamental results have been obtained for the Schrodinger operator L. Among them, for example, are questions about the existence of a resolvent, separability (coercive estimate), various weight estimates, estimates of intermediate derivatives of functions from the domain of definition of an operator, estimates of eigenvalues and singular numbers (s-numbers). At present, there are various generalizations of the above results for elliptic operators. For general differential operators, the solution of such problem as a whole is far from complete. In particular, as far as we know, there was no result until now showing the existence of the resolvent and coercivity, as well as the discreteness of the spectrum of a hyperbolic type operator in an infinite domain with increasing and oscillating coefficients. It is easy to see that the study of some classes of differential operators of hyperbolic type defined in the space L2(Rn+1), using the Fourier method, can be reduced to the study of the Schrodinger operator with a negative parameter : Lt = —Д + (—t2 + itb(x) + q(x)), where t is a parameter (—to < t < TO), i2 = —1. Hence, it is easy to see that we get —t2 ^ —to when |t| ^ to for the operator Lt. Consequently, a completely different situation arises here compared to the Schrodinger operator L = — Д + q(x), and in particular, the methods worked out for the Schroodinger operator L turn out to be little adapted when studying the Schrodinger operator Lt with a negative parameter. All these questions indicate the relevance and novelty of this work. In the paper we study the problems of the existence of the resolvent and the coercivity of the Schroodinger operator with a negative parameter. Keywords: Schroodinger operator, singular differential operator, hyperbolic type, negative parameter, coercive estimates, resolvent.