Time evolution of superoscillations for the Schrödinger equation on $${\mathbb {R}}\setminus \{0\}$$

Abstract

AbstractIn quantum mechanics, superoscillations, or the more general supershifts, appear as initial conditions of the time-dependent Schrödinger equation. Already in [5], a unified approach was developed, which yields time persistence of the supershift property under certain holomorphicity and growth assumptions on the corresponding Green’s function. While that theory considers the Schrödinger equation on the whole real line $${\mathbb {R}}$$ R , this paper takes the natural next step and considers $$\mathbb {R}\setminus \{0\}$$ R \ { 0 } , while allowing boundary conditions at $$x=0^\pm $$ x = 0 ± . In particular, the singular $$\frac{1}{x^2}$$ 1 x 2 -potential as well as the very important $$\delta $$ δ and $$\delta '$$ δ ′ distributional potentials are covered.