×

On antisymmetric quasimodes of the Schrödinger operator in the \(n\)-particle problem. (English. Russian original) Zbl 1132.81022

Differ. Equ. 42, No. 4, 576-585 (2006); translation from Differ. Uravn. 42, No. 4, 540-548 (2006).
The author considers the spectral problem \[ H_N\Psi_N(x_1,\dots,x_N;h) = E\Psi_N(x_1,\dots,x_N;h), \Psi_N\in L^2(({\mathbb R}^d)^{\times N}), \] where \(\Psi_N\) is antisymmetric with respect to permutations of \(x_1,x_2,\dots,x_N\), \(H_N\) is a Hermitian \(h\)-pseudodifferential operator in \(L^2(({\mathbb R}^d)^{\times N})\) (quantum systems of \(N\) identical fermions). It is assumed the existence of invariant tori for the one-particle Hamiltonian flow. Under additional geometrical conditions, the author constructs corresponding quasimodes. The structure of these quasimodes is also investigated.

MSC:

81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Karasev, M.V. and Maslov, V.P., Nelineinye skobki Puassona. Geometriya i kvantovanie (Nonlinear Poisson Brackets. Geometry and Quantization), Moscow: Nauka, 1991. · Zbl 0731.58002
[2] Maslov, V.P., Mat. Zametki, 1997, vol. 62, no. 4, pp. 633–634.
[3] Ruuge, A.E., On the Structure of the Ground Antisymmetric State of a Semiclassical Series of the Schrödinger Operator, Preprint MSU, Moscow: Moscow State Univ., 1999, no. 4.
[4] Ruuge, A.E., Mat. Zametki, 2000, vol. 67, no. 2, pp. 257–269.
[5] Ruuge, A.E., Mat. Zametki, 1999, vol. 66, no. 1, pp. 154–156.
[6] Maslov, V.P., Kompleksnyi metod VKB v nelineinykh uravneniyakh (The Complex WKB Method in Nonlinear Equations), Moscow: Nauka, 1977. · Zbl 0449.58001
[7] Arnol’d, V.I., Matematicheskie metody klassicheskoi mekhaniki (Mathematical Methods of Classical Mechanics), Moscow: Nauka, 1989.
[8] Arnol’d, V.I., Funktsional. Anal. i Prilozhen., 1972, vol. 6, no. 2, pp. 12–20.
[9] Johnson, R.A. and Sell, G.R., J. Differential Equations, 1981, vol. 41, pp. 262–288. · Zbl 0443.34037
[10] Dobrokhotov, S.Yu. and Shafarevich, A.I., in Topologicheskie metody v teorii gamil’tonovykh sistem (Topological Methods in the Theory of Hamiltonian Systems), Bolsinov, A.V., Fomenko, A.T., and Shafarevich, A.I., Eds., Moscow, 1998, pp. 41–114.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.