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In this paper we study the time dependent Schrodinger equation with all possible self-adjoint singular interactions located at the origin, which include the $delta$ and $delta$-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Greens 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 Greens function we study the stability and oscillatory properties of the solution of the Schrodinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time dependent Schrodinger equation is an important and challenging problem in quantum mechanics and mathematica
We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the solitonic regio
A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this
In a previous work [Andrade textit{et al.}, Phys. Rep. textbf{647}, 1 (2016)], it was shown that the exact Greens function (GF) for an arbitrarily large (although finite) quantum graph is given as a sum over scattering paths, where local quantum effe
We consider random elliptic equations in dimension $dgeq 3$ at small ellipticity contrast. We derive the large-distance asymptotic expansion of the annealed Greens function up to order $4$ in $d=3$ and up to order $d+2$ for $dgeq 4$. We also derive a