Abstract
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time-dependent Schrödinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper, we give a unified approach to determine the supershift property for the solution of the time-dependent one-dimensional Schrödinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Green’s function, but not its explicit form. With this efficient general technique, we are able to treat various potentials.
Highlights
Superoscillations are band limited functions F that oscillate faster than their fastest Fourier component; they appear in connection with weak measurements in quantum mechanics [2,15,19,31] and as initial conditions in the time-dependent onedimensional Schrödinger equation
For a Green’s function satisfying Assumption 3.1 and an exponentially bounded holomorphic initial condition F, we show in Theorem 3.4 that (1.4) can be viewed as (t, x) = lim e−εy2 G(t, x, y)F(y)dy = eiα G(t, x, yeiα)F(yeiα)dy ε→0+ R
Afterwards, we prove in Theorem 4.5 the time persistence of the supershift property for potentials V, where the corresponding Green’s function satisfies Assumption 3.1
Summary
Superoscillations are band limited functions F that oscillate faster than their fastest Fourier component; they appear in connection with weak measurements in quantum mechanics [2,15,19,31] and as initial conditions in the time-dependent onedimensional Schrödinger equation. (0, x) = F(x), or in optics, signal processing, and other fields of physics and engineering as, e.g. antenna theory [25,43]. The theory of superoscillations and their applications has grown enormously in the last decades, and without claiming completeness, we mention the contributions [24], [26–30] and [35,37,38,41,42]. The standard example of a sequence of superoscillating functions is n. Mathematics Subject Classification: 81Q05, 35A08, 32A10 Keywords: Superoscillating function, Supershift property, Green’s function, Schrödinger equation
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