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Variational principle for semiclassical fermions. (English. Russian original) Zbl 1032.81516

Math. Notes 66, No. 1, 120-123 (1999); translation from Mat. Zametki 66, No. 1, 154-156 (1999).
From the text: In [Mat. Zametki 62, 633-634 (1997; Zbl 1031.81518)], V. P. Maslov suggested the following variational principle \[ \delta \int dx_1\cdots ds_N \Psi^*_\Phi(\widehat H- E)\Psi_\Phi= 0,\tag{1} \] where \[ \Psi_\Phi(x_1,\dots, x_N)= \prod_{1\leq i< j\leq N}\Phi(x_i, x_j),\quad \Phi(\xi, x)= -\Phi(x, \xi),\quad x_i\in\mathbb{R}^d;\tag{2} \] he used it to estimate the antisymmetric ground state of the Hamiltonian \[ \widehat H= \sum^N_{i=1} \Biggl(-\Biggl({h^2\over 2}\Biggr) {\partial^2\over\partial x^2_i}+ u(x_i)\Biggr)+ \Biggl({1\over N}\Biggr) \sum_{i< j} V(x_i, x_j). \] In the general case, in which \(d\geq 1\), the variational principle (1), (2) belongs to a rather limited class of problems in which \(h\to 0\) and \(N\to \infty\) (with some relations between \(h\) and \(N\) that are defined differently, depending on the specific form of the Hamiltonian \(\widehat H_N\)). Therefore, it is of interest to state an analog of this variational principle in a wider domain.
In this communication, we state a generalization of (1), (2) corresponding to the following statement of the problem: first we consider the passage to the limit \(h\to 0\) and only then we let the particle number \(N\) tend to infinity. The outline of the arguments leading to this generalization is as follows. First, we consider the semiclassical asymptotics \((h\to 0)\) of the antisymmetric ground state of the Hamiltonian of the problem. It can be expressed in a form similar to (2) by using functions whose number is finite and independent of \(N\). The functions themselves are \(N\)-dependent but have a limit as \(N\to\infty\); this makes it possible to state a generalization of the variational principle (1), (2).

MSC:

81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory

Citations:

Zbl 1031.81518
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References:

[1] V. P. Maslov,Mat. Zametki [Math. Notes],62, No. 4, 633–634 (1997).
[2] V. P. Maslov,Operator Methods [in Russian], Nauka, Moscow (1973).
[3] L. D. Landau and E. M. Lifshits,Quantum Mechanics [in Russian], Fizmatlit, Moscow (1963).
[4] N. N. Bogolyubov, V. V. Tolmachev, and D. V. Shirkov,A New Method in the Theory of Superconductivity [in Russian], Izd. AN SSSR, Moscow (1958). · Zbl 0083.45405
[5] G. V. Koval’,Russ. J. Math. Phys.,5, No. 4, 527–528 (1998).
[6] V. P. Maslov,Complex WKB Method in Nonlinear Equations [in Russian], Nauka, Moscow (1977). · Zbl 0449.58001
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