TY - JOUR
T1 - Green's function for the Schrödinger equation with a generalized point interaction and stability of superoscillations
AU - Aharonov, Yakir
AU - Behrndt, Jussi
AU - Colombo, Fabrizio
AU - Schlosser, Peter
N1 - Publisher Copyright:
© 2020 The Author(s)
PY - 2021/3/15
Y1 - 2021/3/15
N2 - In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the δ and δ′-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.
AB - In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the δ and δ′-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.
KW - Green's function
KW - Point interaction
KW - Schrödinger equation
KW - Superoscillating function
UR - http://www.scopus.com/inward/record.url?scp=85098991810&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2020.12.029
DO - 10.1016/j.jde.2020.12.029
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AN - SCOPUS:85098991810
SN - 0022-0396
VL - 277
SP - 153
EP - 190
JO - Journal of Differential Equations
JF - Journal of Differential Equations
ER -