 04101 HansChristoph Kaiser, Hagen Neidhardt, Joachim Rehberg
 Convexity of trace functionals and Schroedinger operators
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Apr 6, 04

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Abstract. Let H be a semibounded selfadjoint operator on 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+Ue(U)))e(U), where U is a bounded selfadjoint operator and e(U) is a
normalizing real functionthe Fermi levelwhich may be identical zero. If additionally F is continuously differentiable, then the corresponding
trace functional is Frechet differentiable and there is an expression of its gradient in terms of 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
Schroedingertype operator and U a realvalued 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 Schroedinger's operatorwhich has been understood since the late 1980sfollows as a special case.
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