On the Schrödinger equation in \(L^ p\) spaces (Q1910162)

The paper examines the optimality of the Sobolev constants for the Schrödinger equation in \(L^p (T^N)\) where \(T^N\) is the \(N\)-dimensional torus and in \(L^p (] - \pi, \pi [^N)\) with Dirichlet or Neumann boundary conditions. The proofs combine the boundary value theorem for holomorphic semigroups, explicit representation of the semigroups in question, Gaussian estimates on Heat kernels and sharp \(L^p\)-estimates on trigonometric polynomials. The result, for the \(N\)-dimensional torus \(T^N\) is the following: The Schrödinger operator \(i \Delta\) on \(L^p (T^N)\), \(1 \leq p < \infty\), is the generator of an \(\alpha\)-times integrated group if \(\alpha > {N \over 2} |{1 \over 2} - {1 \over p} |\). However, \(iA\) does not generate an \(\alpha\)-times integrated group if \(\alpha < {N \over 2} |{1 \over 2} - {1 \over p} |\).