On persistence of superoscillations for the Schrödinger equation with time‐dependent quadratic Hamiltonians
On persistence of superoscillations for the Schrödinger equation with time‐dependent quadratic Hamiltonians
复制标题
具有时间依赖性二次哈密顿量的薛定谔方程的超振荡持续性
DOI：
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发表时间：
2018
影响因子：
2.9
通讯作者：
E. Suazo
中科院分区：
文献类型：
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作者：
Elijah Hight;T. Oraby;Jose A. A. Palacio;E. Suazo
在这项工作中，我们证明了具有含时系数的薛定谔方程的超振动在时间上的持久性。为了证明超振动的持久性，我们将系数满足Riccati系统，并将解表示为卷积算子，表示为Riccati系统的解。此外，我们还显式地解决了具有三种不同超振荡初值的柯西初值问题。运算符是在整函数空间上定义的。具体的例子包括Caldieroa-Kanai和简并的参数谐振子哈密顿量，而其他例子可能包括非自共轭的哈密顿量。对于这些例子，我们用数值方法说明了实部和虚部的收敛。
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.