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Register a new userComplete ionization for a non-autonomous point interaction model in d = 2
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We consider the two dimensional Schrodinger equation with time dependent delta potential, which represents a model for the dynamics of a quantum particle subject to a point interaction whose strength varies in time. First, we prove global well-posedness of the associated Cauchy problem under general assumptions on the potential and on the initial datum. Then, for a monochromatic periodic potential (which also satisfies a suitable no-resonance condition) we investigate the asymptotic behavior of the survival probability of a bound state of the time-independent problem. Such probability is shown to have a time decay of order $mathcal{O}(t^{-1})$, up to exponentially fast decaying terms.
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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.
We present a new hydrodynamic model for synchronization phenomena which is a type of pressureless Euler system with nonlocal interaction forces. This system can be formally derived from the Kuramoto model with inertia, which is a classical model of interacting phase oscillators widely used to investigate synchronization phenomena, through a kinetic description under the mono-kinetic closure assumption. For the proposed system, we first establish local-in-time existence and uniqueness of classical solutions. For the case of identical natural frequencies, we provide synchronization estimates under suitable assumptions on the initial configurations. We also analyze critical thresholds leading to finite-time blow-up or global-in-time existence of classical solutions. In particular, our proposed model exhibits the finite-time blow-up phenomenon, which is not observed in the classical Kuramoto models, even with a smooth distribution function for natural frequencies. Finally, we numerically investigate synchronization, finite-time blow-up, phase transitions, and hysteresis phenomena.
We review previous work on spectral flow in connection with certain self-adjoint model operators ${A(t)}_{tin mathbb{R}}$ on a Hilbert space $mathcal{H}$, joining endpoints $A_pm$, and the index of the operator $D_{A}^{}= (d/d t) + A$ acting in $L^2(mathbb{R}; mathcal{H})$, where $A$ denotes the operator of multiplication $(A f)(t) = A(t)f(t)$. In this article we review what is known when these operators have some essential spectrum and describe some new results in terms of associated spectral shift functions. We are especially interested in extensions to non-Fredholm situations, replacing the Fredholm index by the Witten index, and use a particular $(1+1)$-dimensional model setup to illustrate our approach based on spectral shift functions.
In this paper, we consider the mean field limit of Brownian particles with Coulomb interaction in 3D space. In particular, using a symmetrization technique, we show that the limit measure almost surely is a weak solution to the limiting nonlinear Fokker-Planck equation. By proving that the energy almost surely is bounded by the initial energy, we improve the regularity of the weak solutions. Moreover, by a natural assumption, we establish the weak strong uniqueness principle, which is closely related to the propagation of chaos.
We prove an analogue of Shnirelman, Zelditch and Colin de Verdieres Quantum Ergodicity Theorems in a case where there is no underlying classical ergodicity. The system we consider is the Laplacian with a delta potential on the square torus. There are two types of wave functions: old eigenfunctions of the Laplacian, which are not affected by the scatterer, and new eigenfunctions which have a logarithmic singularity at the position of the scatterer. We prove that a full density subsequence of the new eigenfunctions equidistribute in phase space. Our estimates are uniform with respect to the coupling parameter, in particular the equidistribution holds for both the weak and strong coupling quantizations of the point scatterer.
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