Sobolev-like cones of trace-class operators on unbounded domains: Interpolation inequalities and compactness properties

Abstract:

In this paper we extend the compactness properties for trace-class operators obtained by Dolbeault, Felmer and Mayorga-Zambrano to a smooth unbounded domain Ω ⊆ ℝ^d, d≥3. We consider V, a non-negative potential on Ω that blows up at infinity, and the normed space H<inf>V</inf>(Ω)={u∈H<inf>0</inf>¹(Ω): ∥u∥<inf>V</inf>²=∫<inf>Ω</inf>(|∇u(x)| ²+|u(x)|²V(x))dx<∞}. A positive self-adjoint trace-class operator R belongs to the Sobolev-like cone ℋ <inf>V,+</inf>¹ if (ψ<inf>i,R</inf>)<inf>ℕ</inf>⊆H <inf>V</inf>(Ω) and 〈 〈〈R〉〉<inf>V</inf>= Σ<inf>i=1</inf>^∞ν<inf>i,R</inf>∥ψ <inf>i,R</inf>∥<inf>V</inf>²< ∞, where (ν<inf>i,R</inf>)<inf>i ∈ ℕ</inf> is the sequence of occupation numbers of R and (ψ<inf>i, R</inf>)<inf>∈ ℕ</inf>⊆L ²(Ω) is a corresponding Hilbertian basis of eigenfunctions. We prove that a sequence in ℋ<inf>V,+</inf>¹, bounded in energy 〈〈·〉〉<inf>V</inf>, has a subsequence that converges in trace norm; this is analogous to the classical Sobolev immersion H ¹(Ω) ⊆ L²(Ω). We prove the existence of lower bounds for nonlinear free energy functionals and, by doing so, we establish Lieb-Thirring type inequalities as well as some Gagliardo-Nirenberg type interpolation inequalities; then our compactness result is applied to minimize nonlinear free energy functionals working on H<inf>V,+</inf> ¹. © 2013 Elsevier Ltd. All rights reserved.

Año de publicación:

2013

Keywords:

• Schrödinger operator
• Trace-class operator
• compactness
• Free energy

Fuente:

scopus