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On persistence of superoscillations for the Schrödinger equation with time‐dependent quadratic Hamiltonians
Author(s) -
Hight Elijah,
Oraby Tamer,
Palacio Jose,
Suazo Erwin
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5992
Subject(s) - mathematics , initial value problem , operator (biology) , riccati equation , quadratic equation , degenerate energy levels , harmonic oscillator , mathematical analysis , schrödinger equation , partial differential equation , quantum mechanics , biochemistry , chemistry , physics , geometry , repressor , transcription factor , gene
In this work, we prove the persistence in time of superoscillations for the Schrödinger equation with time‐dependent coefficients. In order to prove the persistence of superoscillations, we have conditioned the coefficients to satisfy a Riccati system, and we have expressed the solution as a convolution operator in terms of solutions of this Riccati system. Further, we have solved explicitly the Cauchy initial value problem with three different kinds of superoscillatory initial data. The operator is defined on a space of entire functions. Particular examples include Caldirola‐Kanai and degenerate parametric harmonic oscillator Hamiltonians, and other examples could include Hamiltonians not self‐adjoint. For these examples, we have illustrated numerically the convergence on real and imaginary parts.