The Lp−Lp′ estimate for the Schrödinger equation on the half-line (original) (raw)

This paper investigates the Lp−Lp′ estimate for the Schrödinger equation defined on the half-line, focusing on the initial-boundary value problem governed by a self-adjoint Hamiltonian with homogeneous Dirichlet boundary conditions. It establishes Lp−Lp′ estimates for the evolution operator associated with the Hamiltonian, particularly under conditions where the Hamiltonian has eigenvalues. By using techniques such as the Fourier transform and properties of Sobolev spaces, the results extend from cases without eigenvalues to those with restricted subspaces orthogonal to these eigenvalues, providing insights into the behavior of solutions over time.