Eigenvalue estimates for Laplacians on measure spaces

Estimates for sums and gaps of eigenvalues of Laplacians on measure spaces

“…This is the main motivation of the present paper. This paper is also a continuation of the work by the authors [8] and by Pinasco and Scarola [32] on estimating the first eigenvalue of Laplacians with respect to fractal measures.…”

Krein-Feller operators on Riemannian manifolds and Minkowski spaces

“…These Laplacians, as well as their generalizations, have been studied extensively in connection with fractal geometry, such as existence of an orthonormal basis of eigenfunctions, spectral dimension and spectral asymptotics, eigenvalues and eigenfunctions, eigenvalue estimates, differential equations, nodal inverse problems, wave equations and wave speed, heat equation and heat kernel estimates, etc. (see [4,7,12,16,[20][21][22][23][24][25]32,35,36,50,51,[53][54][55][56][57]62,63,69] and references therein). We remark that Freiberg and her coauthors defined a class of Laplacians that are more general in that Lebesgue measure is replaced by a more general measure (see, e.g., [21,23]).…”

Spectral asymptotics of Laplacians associated with a class of higher-dimensional graph-directed self-similar measures *

“…These Laplacians are also known as Kreȋn-Feller operators; they were first studied by Kreȋn and Feller in the 1950s [8,9,17,18]. In the fractal setting they have been studied quite extensively, including properties of the Laplacian [12,16], eigenvalues and eigenfunctions [3,6], spectral asymptotics [10,13,28,30,32,34,37], eigenvalue estimates [7,35], wave equation and wave speed [1,4,33], heat equation and heat kernel estimates [14,39], Schrödinger operators [22,31], and inverse problems [11].…”