Ensemble Generalized Kohn-Sham Theory: The Good, the Bad, and the Ugly

Two important extensions of Kohn-Sham (KS) theory are generalized KS theory and ensemble KS theory. The former allows for non-multiplicative potential operators and greatly facilitates practical calculations with advanced, orbital-dependent functionals. The latter allows for quantum ensembles and enables the treatment of, e.g., open systems and excited states. Here, we combine the two extensions, both formally and practically, first via an exact yet complicated formalism, then via a computationally tractable variant that involves a controlled approximation of ensemble "ghost interactions" by means of an iterative algorithm. The resulting formalism is illustrated using selected examples. This opens the door to the application of generalized KS theory in more challenging quantum scenarios and to the improvement of ensemble theories for the purpose of practical and accurate calculations.<br><br>

A saddle point type solution for a system of operator equations

Let Ω⊂Rn n>1 and let p,q≥2. We consider the system of nonlinear Dirichlet problems equation* brace(Au)(x)=Nu′(x,u(x),v(x)),x∈Ω,r-(Bv)(x)=Nv′(x,u(x),v(x)),x∈Ω,ru(x)=0,x∈∂Ω,rv(x)=0,x∈∂Ω,endequation* where N:R×R→R is C1 and is partially convex-concave and A:W01,p(Ω)→(W01,p(Ω))* B:W01,p(Ω)→(W01,p(Ω))* are monotone and potential operators. The solvability of this system is reached via the Ky–Fan minimax theorem.

Parameterized splitting theorems and bifurcations for potential operators, Part II: Applications to quasi-linear elliptic equations and systems

<p style='text-indent:20px;'>This is the second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the first part we obtain many new bifurcation results for quasi-linear elliptic boundary value problems of higher order.</p>

Parameterized splitting theorems and bifurcations for potential operators, Part I: Abstract theory

<p style='text-indent:20px;'>This is the first part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using parameterized versions of splitting theorems in Morse theory we generalize some famous bifurcation theorems for potential operators by weakening standard assumptions on the differentiability of the involved functionals, which opens up a way of bifurcation studies for quasi-linear elliptic boundary value problems.</p>