A method for calculating the eigenvalue of a many-body system without solvingthe eigenfunction is suggested. In many cases, we only need the knowledge ofeigenvalues rather than eigenfunctions, so we need a method solving only theeigenvalue, leaving alone the eigenfunction. In this paper, the method isestablished based on statistical mechanics. In statistical mechanics,calculating thermodynamic quantities needs only the knowledge of eigenvaluesand, then, the information of eigenvalues is embodied in thermodynamicquantities. The method suggested in the present paper is indeed a method forextracting the eigenvalue from thermodynamic quantities. As applications, wecalculate the eigenvalues for some many-body systems. Especially, the method isused to calculate the quantum exchange energies in quantum many-body systems.Moreover, using the method we calculate the influence of the topological effecton eigenvalues.
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