Convexity of trace functionals and Schroedinger operators
Please always quote using this URN:urn:nbn:de:0296-matheon-500
- Let H be a semi–bounded self–adjoint operator in a separable Hilbert space. For a certain class of positive, continuous, decreasing, and convex functions F we show the convexity of trace functionals tr(F (H + U − ε(U ))) − ε(U ), where U is a bounded self–adjoint operator on H and ε(U ) is a normalizing real function—the Fermi level—which may be identical zero. If additionally F is continuously differentiable, then the corresponding trace functional is Fréchet differentiable and there is an expression of its gradient in terms off the derivative of F . The proof of the differentiability of the trace functional is based upon Birman and Solomyak’s theory of double Stieltjes operator integrals. If, in particular, H is a Schrödinger–type operator and U a real-valued function, then the gradient of the trace functional is the quantum mechanical expression of the particle density with respect to an equilibrium distribution function f = −F . Thus, the monotonicity of the particle density in its dependence on the potential U of Schrödinger’s operator—which has been understood since the late 1980s—follows as a special case.
Author: | Hans-Christoph Kaiser, Hagen Neidhardt, Joachim Rehberg |
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URN: | urn:nbn:de:0296-matheon-500 |
Referee: | Anton Bovier |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2004/01/21 |
Release Date: | 2004/01/20 |
Institute: | Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS) |
Preprint Number: | 46 |