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Convexity of trace functionals and Schrödinger operators by Hans-Christoph Kaiser; Hagen Neidhardt; Joachim Rehberg is a Mathematics article available to read on EtoBox.

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Let H be a semi-bounded self-adjoint 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 of the form tr(F (H + Uε(U )))ε(U ), where U is a bounded, self-adjoint operator 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 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 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.

Who reads Convexity of trace functionals and Schrödinger operators?

Author

Hans-Christoph Kaiser; Hagen Neidhardt; Joachim Rehberg

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0022-1236)

Published

2006

Language

EN

Field

Mathematics (Physical Sciences)