Eigenvalue Approximation for Krein-Feller-Operators

Abstract

We study the limiting behavior of the eigenvalues of Krein-Feller-Operators with respect to weakly convergent probability measures. Therefore, we give a representation of the eigenvalues as zeros of measure theoretic sine functions. Further, we make a proposition about the limiting behavior of the previously determined eigenfunctions.With the main results we finally determine the speed of convergence of eigenvalues and -functions for sequences which converge to invariant measures on the Cantor set.

Spectral asymptotics of Krein--Feller operators for weak Gibbs measures on self-conformal fractals with overlaps

“…We investigate the spectral properties of the classical Kreȋn-Feller operator ∆ for weak Gibbs measures on self-conformal fractals contained in the open unit interval with and without overlaps (see 4) under Dirichlet boundary conditions. Spectral properties of the operator ∆ have attracted much attention in the last century, beginning with Feller [Fel57], Kac [Kac59], Hong and Uno [UH59], McKean and Ray [MR62], Kotani and Watanabe [KW82], Fujita [Fuj87], Solomyak and Verbitsky [SV95] and more recently by Vladimirov and Sheȋpak [VS13], Faggionato [Fag12], Arzt [Arz14;Arz15], Ngai [Nga11], Ngai, Tang and Xie [NTX18; NX20] and Freiberg, Minorics [Min20;FM19].…”

An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians

“…First, we give convergence results for the generalized hyperbolic functions introduced in Section 3 using results from [20]. Let p k , q k , k ∈ N be defined by µ and p k,n , q k,n , k ∈ N be defined by µ n for n ∈ N. For z ∈ R let cosh z , sinh z be defined by µ and cosh z,n , sinh z,n be defined by µ n for n ∈ N. We obtain a result for the generalized hyperbolic functions, comparable to that for the trigonometric functions in [20].…”

“…µ is a so-called Cantor measure. Following [20], for n ∈ N we define the approximating Cantor measures of level n by Finally, we connect both applications. Example 6.3: Let ε > 0, n ∈ N and let µ, µ n , {u(t) : t ≥ 0} and {u n (t) : t ≥ 0} be defined as in Example 6.2.…”

“…Krein-Feller operators defined by measures on the classic Cantor set or, more general, Cantor-like sets with gaps have been extensively studied in recent years (see e.g. [2,[17][18][19][20]). In this paper, we give an interpretation of a solution to (2) in the case where µ is not of full support.…”