Spectral Asymptotics for Krein-Feller-Operators with respect to Random Recursive Cantor Measures

Spectral asymptotics for Krein–Feller operators with respect to 𝑉-variable Cantor measures

“…For the recursive case, we take almost the same setting with the only difference that the index set J has not to be countable. As in [19], we let J be an index set and as before we define to each j ∈ J an iterated function system S (j) = S (j) 1 , ..., S (j)…”

“…N/D and θ i ω if we mean the sub tree θ i I(ω) of I(ω) which is rooted at i ∈ I(ω). Under some regularity conditions, which are basically conditions on the underlying (C-M-J) Branching process (for reference see [19]), we receive the following theorem.…”

“…With these notations, we can easily transfer the definition of recursive Cantor sets (see e.g. [10] or [19]) to V -variable Cantor sets:…”

“…The spectral asymptotics for Krein-Feller-operators with respect to self similar measures was developed by Freiberg [7], with respect to random (and deterministic) homogeneous Cantor measures by Arzt [1] and w.r.t. random recursive Cantor measures in [19].…”

Eigenvalue Approximation for Krein-Feller-Operators

“…Analytic properties of Krein-Feller-Operators are developed in [7]. Many papers deal with this operator and with the resulting eigenvalue problem, see for example Feller [6], Freiberg et al [7][8][9][10][11][12][13]15], Fujita [16], Minorics [20,21], Ngai et al [3,4,17,23,24] and for higher dimensional generalizations Freiberg and Seifert [14] and Solomyak et al [22,26].…”