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Register a new userAnalysis of the Toolkit method for the time-dependant Schrodinger equation
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Authors
Lucie Baudouin
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The goal of this paper is to provide an analysis of the toolkit method used in the numerical approximation of the time-dependent Schrodinger equation. The toolkit method is based on precomputation of elementary propagators and was seen to be very efficient in the optimal control framework. Our analysis shows that this method provides better results than the second order Strang operator splitting. In addition, we present two improvements of the method in the limit of low and large intensity control fields.
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We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrodinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrodinger kind.
The three-dimensional Schrodinger equation with a position-dependent (effective) mass is studied in the framework of Supersymmetrical (SUSY) Quantum Mechanics. The general solution of SUSY intertwining relations with first order supercharges is obtained without any preliminary constraints. Several forms of coefficient functions of the supercharges are investigated and analytical expressions for the mass function and partner potentials are found. As usual for SUSY Quantum Mechanics with nonsingular superpotentials, the spectra of intertwined Hamiltonians coincide up to zero modes of supercharges, and the corresponding wave functions are connected by intertwining relations. All models are partially integrable by construction: each of them has at least one second order symmetry operator.
In this work, we study the Schrodinger equation $ipartial_tpsi=-Deltapsi+eta(t)sum_{j=1}^Jdelta_{x=a_j(t)}psi$ on $L^2((0,1),C)$ where $eta:[0,T]longrightarrow R^+$ and $a_j:[0,T]longrightarrow (0,1)$, $j=1,...,J$. We show how to permute the energy associated to different eigenmodes of the Schrodinger equation via suitable choice of the functions $eta$ and $a_j$. To the purpose, we mime the control processes introduced in [17] for a very similar equation where the Dirac potential is replaced by a smooth approximation supported in a neighborhood of $x=a(t)$. We also propose a Galerkin approximation that we prove to be convergent and illustrate the control process with some numerical simulations.
Recently two approximate Newton methods were proposed for the optimisation of Markov Decision Processes. While these methods were shown to have desirable properties, such as a guarantee that the preconditioner is negative-semidefinite when the policy is $log$-concave with respect to the policy parameters, and were demonstrated to have strong empirical performance in challenging domains, such as the game of Tetris, no convergence analysis was provided. The purpose of this paper is to provide such an analysis. We start by providing a detailed analysis of the Hessian of a Markov Decision Process, which is formed of a negative-semidefinite component, a positive-semidefinite component and a remainder term. The first part of our analysis details how the negative-semidefinite and positive-semidefinite components relate to each other, and how these two terms contribute to the Hessian. The next part of our analysis shows that under certain conditions, relating to the richness of the policy class, the remainder term in the Hessian vanishes in the vicinity of a local optimum. Finally, we bound the behaviour of this remainder term in terms of the mixing time of the Markov chain induced by the policy parameters, where this part of the analysis is applicable over the entire parameter space. Given this analysis of the Hessian we then provide our local convergence analysis of the approximate Newton framework.
In this paper we prove an approximate controllability result for the bilinear Schrodinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrodinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite dimensional Galerkin approximations and allows to get estimates for the $L^{1}$ norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields.
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