Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field

Dobrokhotov, S. Yu.; Anikin, A. Yu.

doi:10.1002/9781118577608.ch5

Cited by 9 publications

(2 citation statements)

References 29 publications

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“…By this transformation the Hamiltonian remains real and gauge invariant. This is consistent with the following example of operator on L 2 (R 2 ) H(y, hD y ) = (hD y1 ) 2 +(hD y2 −by 1 ) 2 +V 1 (y 1 )+ω 2 y 2 2 (10) which is unitarily equivalent (after partial Fourier transform with respect to x 2 ) to the Schrödinger operator without a magnetic field [2], [6]). It turns out that our computations lead (up to the present accuracy) to the same quantities as [10], but without factor i at some places, see e.g.…”

Section: Geometry and Quantization Of Magnetic Hamiltonianssupporting

confidence: 86%

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On semi-classical spectral series for an atom in a periodic polarized electric field

Ifa¹,

Louati²,

Rouleux³

2021

Preprint

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In this report we present preliminary results about the tunneling problem for a magnetic Schrödinger operator. As a motivation we consider the 3-D time-dependent Schrödinger operatorwhere V is a radial potential and E(t) a circularly polarized field with uniform frequency ω.The quantum monodromy operator (QMO) that takes the system through a complete period T = 2π/ω, turns out to be unitarily equivalent to e iT P A (x,hDx)/h , where P A (x, hDx)) identifies with a magnetic Schrödinger operator. When V is sufficiently confining, P A (x, hDx)) presents a double magnetic well. Then we construct its semi-classical ground state and examine the splitting between its two first eigenvalues.

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Section: Geometry and Quantization Of Magnetic Hamiltonianssupporting

confidence: 86%

“…So we have to compute w near minimal geodesics γ E ′ (h) between ∂U ± E ′ (h) . Such (finitely many) minimal geodesics are also called librations [2], [6], [1]. Within the required accuracy on tunneling rates, we could again replace E ′ (h) by E ′ , which amounts to replace the librations by the instanton between U ± E ′ = {y ± 0 }.…”

mentioning

confidence: 99%