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On the spectrum of the one-particle density matrix. (English. Russian original) Zbl 1479.81009

Funct. Anal. Appl. 55, No. 2, 113-121 (2021); translation from Funkts. Anal. Prilozh. 55, No. 2, 44-54 (2021).
Summary: The one-particle density matrix \(\gamma(x, y)\) is one of the key objects in quantum-mechanical approximation schemes. The self-adjoint operator \(\Gamma\) with kernel \(\gamma(x, y)\) is trace class, but no sharp results on the decay of its eigenvalues were previously known. The note presents the asymptotic formula \(\lambda_k \sim (Ak)^{-8/3}\), \(A \ge 0\), as \(k\to\infty\) for the eigenvalues \(\lambda_k\) of the operator \(\Gamma\) and describes the main ideas of the proof.

MSC:

81P15 Quantum measurement theory, state operations, state preparations
81Q15 Perturbation theories for operators and differential equations in quantum theory
47L07 Convex sets and cones of operators
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