Complex powers of the Schrödinger operator with a regularly growing potential
Let \(A\) be the Schrödinger operator \[ Au(x)= -\Delta u(x)+ Q(x) u(x), \qquad x\in \mathbb{R}^ m, \] where the potential \(Q\) is assumed to be a nonnegative, infinitely differentiable and regularly increasing function at infinity, i.e., \[ c| x^ \alpha |\leq Q(x)\leq C| x|^ \alpha, \qquad \alpha>0, \] for \(| x|\) sufficiently large. Using the well established theory of pseudo-differential operators [see \textit{L. Hörmander}, Commun. Pure Appl. Math. 32, 359-443 (1979; Zbl 0388.47032)], the author finds the parametrix -- asymptotic expansion of the resolvent \((A- \lambda)^{-1}\) and then obtains the representation of the complex powers \(A^ z\) of the operators \(A\). In particular, he proves that the function \(z\mapsto A^ z\) can be meromorphically continued to the whole complex plain with poles of order not greater than two.
scientific article; zbMATH DE number 165458