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A theorem on the classical limit as \(\hbar\to 0\) for a system of \(N\) interacting fermions. (English. Russian original) Zbl 0924.35124

Math. Notes 63, No. 1, 131-132 (1998); translation from Mat. Zametki 63, No. 1, 145-146 (1998).
The author considers the Schrödinger equation for \(N\) interacting identical particles and compares the classical limit \(\hbar\rightarrow 0\) with the Liouville equation. Two (apparently new) theorems are presented in this very short paper. According to the first, Wigner distributions for pairs of solutions converge (weakly) to generalized solutions of the Liouville equation, provided the initial data of the Wigner distributions converge. The second theorem compares eigenvalues of the Liouville operator with differences of eigenvalues of the Hamilton operator to first order in \(\hbar\).

MSC:

35Q40 PDEs in connection with quantum mechanics
81V70 Many-body theory; quantum Hall effect
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References:

[1] V. P. Maslov,Mat. Zametki [Math. Notes],62, No. 5, 633–634 (1997).
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