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arXiv:1411.4112 · Journal of Mathematical Physics

Quantum harmonic oscillator with superoscillating initial datum

Roman V. Buniy, Fabrizio Colombo, Irene Sabadini, Daniele Carlo Struppa

Published Nov 1, 2014• 44 citations
arXivHTMLPDF

Abstract

In this paper, we study the evolution of superoscillating initial data for the quantum driven harmonic oscillator. Our main result shows that superoscillations are amplified by the harmonic potential and that the analytic solution develops a singularity in finite time. We also show that for a large class of solutions of the Schrödinger equation, superoscillating behavior at any given time implies superoscillating behavior at any other time.

DOI
10.1063/1.4901753

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In this paper, we study the evolution of superoscillating initial data for the quantum driven harmonic oscillator. Our main result shows that superoscillations are amplified by the harmonic potential and that the analytic solution develops a singularity in finite time. We also show that for a large class of solutions of the Schrödinger equation, superoscillating behavior at any given time implies superoscillating behavior at any other time.
Weak measurements and the theory of weak values have a very deep meaning in quantum mechanics, and new phenomena associated with them has recently been observed experimentally. This theory has also directly led to the notion of superoscillating sequences of functions. In this paper we consider Cauchy problems with superoscillatory initial conditions (in particular, the Cauchy problem for the Schrödinger equation and some of its variations), and we give conditions under which the superoscillations persist in time. Our work is based on results from the theory of formal solutions to Cauchy problems and on the study of the specific growth of superoscillatory sequences, when regarded as functions of a complex variable. There are two main aims of this paper: one is to explain the mathematical tools that are necessary to study superoscillations, also repeating a few results that we have already proved in other papers in order to clarify the strategy. The second aim is to show that our technique applies to a large class of problems, showing under which conditions the superoscillatory phenomenon persists. Finally, we point out that our strategy can be applied also to non-constant coefficients differential equations as the quantum harmonic oscillator.
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.
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.
Time evolution of superoscillations for the Schrödinger equation on R∖{0}{\mathbb {R}}\setminus \{0\}R∖{0}
Abstract In 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 R{\mathbb {R}}R R , this paper takes the natural next step and considers R∖{0}\mathbb {R}\setminus \{0\}R∖{0} R \ { 0 } , while allowing boundary conditions at x=0±x=0^\pm x=0± x = 0 ± . In particular, the singular 1x2\frac{1}{x^2}x21​ 1 x 2 -potential as well as the very important δ\delta δ δ and δ′\delta 'δ′ δ ′ distributional potentials are covered.
In this work we study the persistence in time of superoscillations for the Schrödinger equation with quadratic time-dependent Hamiltonians. We have solved explicitly the Cauchy initial value problem with three different kind of oscillatory initial data. In order to prove the persistence of superoscillations we have defined explicitly an operator in terms of solutions of a Riccati system associated with the variable coefficients of the Hamiltonian. The operator is defined on a space of entire functions. Particular examples include Caldirola-Kanai and degenerate parametric harmonic oscillator Hamiltonians and more. For these examples we have illustrated numerically the convergence on real and imaginary parts.
Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. These functions appear in various fields of science and technology, in particular they were discovered in quantum mechanics in the context of weak values introduced by Y. Aharonov and collaborators. The evolution problem of superoscillatory functions as initial conditions for the Schrödinger equation is intensively studied nowadays and the supershift property of the solution of Schrödinger equation encodes the persistence of superoscillatory phenomenon during the evolution. In this paper, we prove that the evolution of a superoscillatory initial datum for spinning particles in a magnetic field has the supershift property. Our techniques are based on the exact propagator of spinning particles, the associated infinite order differential operators and their continuity on suitable spaces of entire functions with growth conditions.