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arXiv:2008.10935

Eigenvalue estimates for the one-particle density matrix

Alexander V. Sobolev

Published Aug 25, 2020• 4 citations
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Abstract

It is shown that the eigenvalues $λ_k, k=1, 2, \dots,$ of the one-particle density matrix satisfy the bound $λ_k\le C k^{-8/3}$ with a positive constant $C$.

DOI
10.48550/arxiv.2008.10935

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It is shown that the eigenvalues λk,k=1,2,…,λ_k, k=1, 2, \dots,λk​,k=1,2,…, of the one-particle density matrix satisfy the bound λk≤Ck−8/3λ_k\le C k^{-8/3}λk​≤Ck−8/3 with a positive constant CCC.
For a system of NNN identical particles in a pure bound state, the density matrices of positive orders ppp and $N\ensuremath{-}p$ have the same nonzero eigenvalues with the same multiplicities. If the number of nonzero eigenvalues is finite, these density matrices are unitarily equivalent.
We obtain bounds for all derivatives of the non-relativistic Coulombic one-particle density matrix γ(x,y)γ(x, y)γ(x,y) near the diagonal x=yx = yx=y.
The internal one-particle density matrix is discussed for Bose-Einstein condensates with a finite number of particles in a harmonic trap. It is found that the outcome of the diagonalization of the single-particle density matrix depends on the choice of the internal coordinates: The Pethick-Pitaevskii (PP) type of internal density matrix, whose analytical eigenvalues and eigenfunctions are evaluated, yields a fragmented condensate, while the Jacobi type of internal density matrix leads to an ideal condensate. We give a criterion for the choice of the internal coordinates: In the macroscopic limit the internal density matrix should have the same eigenvalues and eigenfunctions as those of the corresponding ideal Bose-Einstein condensate in the laboratory frame, this being a very physical condition. One choice satisfying this boundary condition is given by the internal Jacobi coordinates, while the internal coordinates with respect to the center of mass of the PP density matrix do not satisfy this condition. Based on our criterion, a general definition of the internal one-particle density matrix is presented in a self-bound system consisting of interacting bosons.
Functions which can be expressed as an antisymmetrized power of a single two-particle function (a geminal) occur in the BCS ansatz. They constitute a comparatively tractable generalization of a single Slater determinant. The second-order reduced density matrix and bounds on its eigenvalues are obtained for the most general such antisymmetrized geminal power (AGP). These eigenvalues are physically of great importance, being analogs, in an analysis of the wavefunction into pair states, of the one-particle occupation numbers of the independent-particle model. A necessary and sufficient condition is given that an alleged first-order reduced density matrix be derivable from an N-particle AGP function. It is shown that only for exceptional functions (called extreme) is the generating geminal of the AGP function an eigenfunction of the 2-matrix. The extreme 2-matrix is diagonalized explicitly and provides a vivid example of the fact that one-particle occupation numbers alone are unable to convey certain information of decisive significance about the wavefunction. The extreme 2-matrix satisfies the assumption basic to current theories of superconductivity.