Besov Spaces for the Schrödinger Operator with Barrier Potential

Abstract

Let H = −d 2/dx 2 + V be a Schrodinger operator on the real line, where $${V=c\chi_{[a,b]}}$$ , c > 0. We define the Besov spaces for H by developing the associated Littlewood–Paley theory. This theory depends on the decay estimates of the spectral operator $${{\varphi}_j(H)}$$ for the high and low energies. We also prove a Mihlin multiplier theorem on these spaces, including the L p boundedness result. Our approach has potential applications to other Schrodinger operators with short-range potentials.