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A tight nonlinear approximation theory for time dependent closed quantum systems

Joseph W. Jerome

Abstract

The approximation of fixed points by numerical fixed points was presented in the elegant monograph of Krasnosel’skii et al. (1972). The theory, both in its formulation and implementation, requires a differential operator calculus, so that its actual application has been selective. The writer and Kerkhoven demonstrated this for the semiconductor drift-diffusion model in 1991. In this article, we show that the theory can be applied to time dependent quantum systems on bounded domains, via evolution operators. In addition to the kinetic operator term, the Hamiltonian includes both an external time dependent potential and the classical nonlinear Hartree potential. Our result can be paraphrased as follows: For a sequence of Galerkin subspaces, and the Hamiltonian just described, a uniquely defined sequence of Faedo–Galerkin solutions exists; it converges in Sobolev space, uniformly in time, at the maximal rate given by the projection operators.

JEL Classification: 35Q41; 47D08; 47H09; 47J25; 81Q05

A Notation and norms

We employ complex Hilbert spaces in this article.

L2(Ω)={f=(f1,,fN)T:|fj|2is integrable onΩ}.
(f,g)L2=j=1NΩfj(x)gj(x)¯dx.

However, ∫Ωfg dx is interpreted as

j=1NΩfjgjdx.

For fL2, as just defined, if each component fj satisfies fjH01(Ω; ℂ), we write fH01(Ω; ℂN), or simply, fH01(Ω). The inner product in H01 is

(f,g)H01=(f,g)L2+j=1NΩfj(x)gj(x)¯dx.

Ωf ⋅ ∇g dx is interpreted as

j=1NΩfj(x)gj(x)dx.

Finally, H–1 is defined as the dual of H01, and its properties are discussed at length in [1]. We use the notation 〈f, ζ〉 for the corresponding duality bracket, which here denotes the action of the continuous linear functional f on the test function ζH01. The Banach space C(J; H01) is defined in the traditional manner:

C(J;H01)={u:JH01:u()is continuous}
uC(J;H01)=suptJu(t)H01.

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Received: 2017-10-17
Revised: 2018-01-29
Accepted: 2018-02-01
Published Online: 2019-09-17
Published in Print: 2019-09-25

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