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.

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Spectral asymptotics of a tensor product of compact operators in Hilbert space with known marginal asymptotics is studied. Methods of A. Karol', A. Nazarov and Ya. Nikitin (Trans. AMS, 2008) are generalized for operators with almost regular marginal asymptotics. In many (but not all) cases it is shown, that tensor product has almost regular asymptotics as well. Obtained results are then applied to the theory of small ball probabilities of Gaussian random fields. §1 IntroductionWe consider compact nonnegative self-adjoint operators T = T * 0 in a Hilbert space H and T in a Hilbert space H. We denote by λ n = λ n (T ) the eigenvalues of the operator T arranged in a nondecreasing order and repeated according to their multiplicity. We also consider their counting functionSimilarly we define λ n and N (t) for T .Having known asymptotics for N (t, T ) and N (t, T ) as t → 0, we aim to determine the asymptotics for N (t, T ⊗ T ). Obtained results are easily generalized to the case of a tensor product of multiple operators.Known applications of such results could be found in problems concerning asymptotics of random values and vectors quantization (see e.g. [1, 2]), average complexity of linear problems, i.e. problems of approximation of a continuous linear operator (see e.g. [3]), and also in the developing theory of small deviations of random processes in L 2 -norm (see e.g. [4,5]).Abstract methods of spectral asymptotics analysis for tensor products, generalized in this paper, were developed in [4] and [5]. In [4] the case is considered, in which the eigenvalues of the operators-multipliers have the so-called regular asymptotic behavior: λ n ∼ ψ(n) n p , n → ∞,

Изучается спектральная асимптотика задачи Штурма-Лиувилля с сингулярной арифметически самоподобной весовой мерой. Полученные ранее результаты А. А. Владимирова и И. А. Шейпака, а также автора, опирающиеся на свойство спектральной периодичности, накладывают значительные ограничения на параметры самоподобия. В данной работе предлагается новый метод оценки считающей функции собственных значений. Это позволяет рассмотреть существенно более широкий класс самоподобных мер.

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