On Spectral Asymptotics of the Neumann Problem for the Sturm–Liouville Equation with Self-Similar Weight of Generalized Cantor Type

Abstract

Spectral asymptotics of the weighted Neumann problem for the Sturm-Liouville equation is considered. The weight is assumed to be the distributional derivative of a self-similar generalized Cantor type function. The spectrum is shown to have a periodicity property for a wide class of Cantor type self-similar functions. A weaker "quasiperiodicity" property is established under certain mixed boundary-value conditions. This allows for a more precise description of the main term of the eigenvalue counting function asymptotics. Previous results by A. A. Vladimirov and I. A. Sheipak are generalized. Bibliography: 17 titles.

On spectral asymptotics of the tensor product of operators with almost regular marginal asymptotics

“…Remark 1. For some Green integral operators with singular arithmetically selfsimilar weight measures (see [16,17,18]) it is shown, that ̺(τ ) is a continuous purely singular function, i.e. its generalized derivative is a measure singular with respect to Lebesgue measure.…”

“…This power asymptotics were considered in the Example 1 and correspond to the case κ 1 = κ 2 = 0. For certain measures µ j functions s j could be constant, but [16,18] describe wide classes of measures, for which the inconstancy of the periodic component is proven.…”

On Spectral Asymptotics of the Sturm–Liouville Problem with Self-Conformal Singular Weight with Strong Bounded Distortion Property

“…where D ∈ (0, 1 2 ) and s is a continuous T -periodic function, dependent on the choice of the weight µ (see also [10] for similar asymptotics in the case of an arbitrary even order differential operator, and [11] for similar results for problems containing two self-similar measures). A series of works [12,13,14] is dedicated to the fine properties of the function s for incrementally generalized classes of self-similar measures.…”

On the spectrum of the Sturm-Liouville problem with arithmetically self-similar weight

“…In the case of singular measure µ it follows from early works by M. G. Krein, that the counting function N : (0, +∞) → N of eigenvalues of the problem (1) admits the estimate o(λ 1 2 ) instead of the usual asymptotics N (λ) ∼ Cλ 1 2 in the case of measure containing a regular component. (see, e.g., [9] or [10], and also [11] for similar results for higher ever order operators and better lower bounds for eigenvalues for some special classes of measures).…”

“…In paper [2] the result of [1] is generalized to the wider class of ladders satisfying conditions (5).…”