2017/01/18 by Batu Güneysu, Güneysu, Batu
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Spectral Theory (math.SP) #math-ph #math.DG #math.MP #math.PR #math.SP
paper · pdf · doi:10.48550/arxiv.1701.04998
16 pages; comments are welcome!
arxiv created 2017/01/18 · arxiv updated 2017/01/19
This paper is a contribution to semiclassical analysis for abstract Schrödinger type operators on locally compact spaces: Let X be a metrizable seperable locally compact space, let μ be a Radon measure on X with a full support. Let (t,x,y)↦ p(t,x,y) be a strictly positive pointwise consistent μ-heat kernel, and assume that the generator Hp≥ 0 of the corresponding self-adjoint contraction semigroup in L2(X,μ) induces a regular Dirichlet form. Then, given a function Ψ: (0,1)→ (0,∞) such that the limit limt→ 0+p(t,x,x)Ψ(t) exists for all x∈ X, we prove that for every potential w:X→ ℝ one has limt → 0+ Ψ(t)tr(e -t Hp + w)= ∫ e-w(x) limt → 0+p(t,x,x) Ψ(t) dμ(x)<∞ for the Schrödinger type operator Hp + w, provided w satisfies very mild conditions at ∞, that are essentially only made to guarantee that the sum of quadratic forms Hp + w/t is self-adjoint and bounded from below for small t, and to guarantee that ∫ e-w(x) limt→ 0+p(t,x,x) Ψ(t) dμ(x)<∞. The proof is probabilistic and relies on a principle of not feeling the boundary for p(t,x,x). In particular, this result implies a new semiclassical limit result for partition functions valid on arbitrary connected geodesically complete Riemannian manifolds, and one also recovers a previously established semiclassical limit result for possibly locally infinite connected weighted graphs.