2019/09/19 by Marc Keßeböhmer, Kesseböhmer, Marc, Aljoscha Niemann +5
Computer Science · Mathematics · #35P20 #42B35 #45D05 #47G30 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Probability (math.PR) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1909.08832
openalex publication_date 2019/09/19 · openalex created_date 2021/11/08 · openalex updated_date 2026/08/01
We consider the generalised Krein-Feller operator Δν, μ with respect to compactly supported Borel probability measures μ and ν with the natural restrictions that μ is atomless, the supp(ν)⊆supp(μ) and the atoms of ν are embedded in the supp(μ). We show that the solutions of the eigenvalue problem for Δν, μ can be transferred to the corresponding problem for the classical Krein-Feller operator Δν∘ Fμ-1, Λ with respect to the Lebesgue measure Λ via an isometric isomorphism determined by the distribution function Fμ of μ. In this way, we obtain a new characterisation of the upper spectral dimension and consolidate many known results on the spectral asymptotics of Krein-Feller operators. We also recover known properties of and connections to generalised gap diffusions associated to these operators.