Convexity of trace functionals and Schrödinger operators
- Kaiser, Hans-Christoph
- Neidhardt, Hagen
- Rehberg, Joachim
2010 Mathematics Subject Classification
- 47H05 46T20 47B10 47F05
2008 Physics and Astronomy Classification Scheme
- 31.15.Ew 31.15.Md 05.30.-d
- trace functionals, convexity, monotonicity, double Stieltjes operator integrals, spectral asymptotics, generalized Fermi level, density-functional theory
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 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 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 ƒ = -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.
- J. Funct. Anal., 234 (2006) pp. 45--69.