Spectral and scattering theory of Schrödinger operators related to the stark effect

Avron, J. E.; Herbst, Ira

doi:10.1007/bf01609485

Cited by 196 publications

(139 citation statements)

References 15 publications

(19 reference statements)

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“…For the propagators, the Avron-Herbst formula [18] gives e Àði="ÞsĤ AE 1 ¼ e Àið 2 s 3 =6"Þ e s@ k e Àði=2"Þ½ðk 2 AE2 Þsþ ks 2 :…”

Section: Prl 103 213001 (2009) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 99%

Accurate Prediction of Nonadiabatic Transitions through Avoided Crossings

Betz

Goddard

2009

Phys. Rev. Lett.

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In the case of one nuclear degree of freedom, we derive an explicit formula for the nuclear wave function transmitted through an avoided crossing, and show that it agrees to high accuracy with precise numerical calculations. DOI: 10.1103/PhysRevLett.103.213001 PACS numbers: 31.70.Hq, 31.50.Gh, 33.50.Hv The time-dependent Born-Oppenheimer approximation (BOA) is at the basis of our understanding of dynamics of molecules. The small ratio " 2 of electronic and nuclear mass allows one to replace the electronic degrees of freedom with an effective potential, and to separate the nuclear dynamics according to the electronic energy surfaces. This dramatically reduces the complexity of the problem.Despite its success, situations where the BOA fails are of great interest in quantum chemistry. These occur when electronic energy levels are not well separated for a given nuclear configuration. Important applications include photodissociation of diatomic molecules like NaI [1], or the reception of light in the retina [2]. Two basic types of failure occur: conical crossings as appearing in [2], and avoided crossings. The latter are typical for systems with one nuclear degree of freedom [1,3], and are the topic of this Letter. We give an explicit formula, cf., (11), for the transmitted wave function at a generic avoided crossing, using only data that is local in time and space. An algorithm for calculating this wave function is then straightforward.The importance of nonadiabatic transitions has led to many efforts to understand and predict them. A simplification of the problem is to replace the nuclear degree of freedom by a classical trajectory. It is both ancient [4] and well understood [5,6], and leads to the famous LandauZener (LZ) formula for the transition probabilities between electronic levels. This formula lies at the basis of several surface hopping models, such as [7][8][9]. While these and other [10] trajectory based methods yield reasonably good transition probabilities, their accuracy in predicting the shape of the transmitted wave function is limited [7]. An improvement to the LZ-transition rates, based on the full quantum scattering theory of the problem, is achieved by the Zhu-Nakamura theory [11]. Again, only transition rates are treated, and not the full transmitted wave function. Hagedorn and Joye [12] derive rigorous asymptotic formulas for the transmitted wave function, in the limit of " ! 0. However, these formulas are difficult to apply in practice and are neither local in time nor space.We consider a two level system with a Hamiltonian with one effective degree of freedom: H ¼ Àð1=2Þ" 2 @ 2 x I þ VðxÞ, where VðxÞ ¼ XðxÞ x þ ZðxÞ z þ dðxÞI is the real-symmetric potential energy matrix in the diabatic representation. I is the 2 Â 2 unit matrix, and x , z are the Pauli matrices. Units are such that @ ¼ 1 and the electron mass m el ¼ 1. " 2 is the ratio of electron and reduced nuclear mass.The time-dependent Schrödinger equation is given by i"@ t c ðx; tÞ ¼ Hc ðx; tÞ, with c 2 L 2 ðR; C 2 Þ, in the time ...

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“…For the propagators, the Avron-Herbst formula [18] gives e Àði="ÞsĤ AE 1 ¼ e Àið 2 s 3 =6"Þ e s@ k e Àði=2"Þ½ðk 2 AE2 Þsþ ks 2 :…”

Section: Prl 103 213001 (2009) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 99%

Accurate Prediction of Nonadiabatic Transitions through Avoided Crossings

Betz

Goddard

2009

Phys. Rev. Lett.

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“…Let us compute the integral kernel of the operator ΦR 0 (z)V Φ * . For the resolvent R 0 (z) one has (see [1]):…”

Section: Proof Of Lemma 34mentioning

confidence: 99%

Trace Formulae and High Energy Asymptotics for the Stark Operator

Pushnitski

2003

Communications in Partial Differential Equations

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In L 2 (R 3 ), we consider the unperturbed Stark operator H 0 (i.e., the Schrödinger operator with a linear potential) and its perturbation H = H 0 + V by an infinitely smooth compactly supported potential V . The large energy asymptotic expansion for the modified perturbation determinant for the pair (H 0 , H) is obtained and explicit formulae for the coefficients in this expansion are given. By a standard procedure, this expansion yields trace formulae of the Buslaev-Faddeev type.

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“…A key role has been played by an explicit formula of Avron and Herbst 18 for the operator when Vϭ0, viz.,…”

Section: Constant Electric and Magnetic Fieldsmentioning

confidence: 99%