Besov-Type and Triebel--Lizorkin-Type Spaces Associated with Heat Kernels

Let $(M, \rho,\mu)$ be an RD-space satisfying the non-collapsing condition. In this paper, the authors introduce Besov-type spaces $B_{p,q}^{s,\tau}(M)$ and Triebel--Lizorkin-type spaces $F_{p,q}^{s,\tau}(M)$ associated to a non-negative self-adjoint operator $L$ whose heat kernels satisfy some Gaussian upper bound estimate, H\"older continuity, and the stochastic completeness property. Characterizations of these spaces via Peetre maximal functions and heat kernels are established for full range of indices. Also, frame characterizations of these spaces are given. When $L$ is the Laplacian operator on $\mathbb R^n$, these spaces coincide with the Besov-type and Triebel-Lizorkin-type spaces on $\mathbb R^n$ studied in [Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010]. In the case $\tau=0$ and the smoothness index $s$ is around zero, comparisons of these spaces with the Besov and Triebel--Lizorkin spaces studied in [Abstr. Appl. Anal. 2008, Art. ID 893409, 250 pp] are also presented.Comment: Collect. Math. (to appear

On the Convergence of the Self-Consistent Field Iteration in Kohn-Sham Density Functional Theory

It is well known that the self-consistent field (SCF) iteration for solving the Kohn-Sham (KS) equation often fails to converge, yet there is no clear explanation. In this paper, we investigate the SCF iteration from the perspective of minimizing the corresponding KS total energy functional. By analyzing the second-order Taylor expansion of the KS total energy functional and estimating the relationship between the Hamiltonian and the part of the Hessian which is not used in the SCF iteration, we are able to prove global convergence from an arbitrary initial point and local linear convergence from an initial point sufficiently close to the solution of the KS equation under assumptions that the gap between the occupied states and unoccupied states is sufficiently large and the second-order derivatives of the exchange correlation functional are uniformly bounded from above. Although these conditions are very stringent and are almost never satisfied in reality, our analysis is interesting in the sense that it provides a qualitative prediction of the behavior of the SCF iteration