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Spectral dimension and Bohr’s formula for the Schrödinger operator on unbalanced fractal space
Journal of Physics A: Mathematical and Theoretical. 2015. Vol. 48. No. 39. P. 1–22.
Molchanov S., Chen J., Teplyaev A.
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Language:
English
Keywords: fractalsфракталыSchrodinger operatorоператор ШредингераHausdorff dimensionspectral dimensionспектральная размерностьразмерность Хаусдорфа
Publication based on the results of:
Dmitry Gayfulin, Hauke M., Nonlinearity 2025 Vol. 38 No. 6 Article 065008
Given an irrational number $\alpha$, we study the asymptotic behaviour of
the Sudler product denoted by $P_N(\alpha) =\prod_{r=1}^N 2\lvert \sin \pi r \alpha \rvert$. We show that $\liminf_{N \to \infty} P_N(\alpha) >0$ and $\limsup_{N \to \infty} P_N(\alpha)/N < \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ exceeds 3 only finitely ...
Added: March 19, 2026
Artigiani M., Hubert P., Skripchenko A., Discrete and Continuous Dynamical Systems 2026 Vol. 47 P. 519–547
We study a class of interval translation mappings introduced by Bruin and Troubetzkoy, describing a new renormalization scheme, inspired by the classical Rauzy induction, for this class. We construct a measure, invariant under the renormalization, supported on the parameters yielding infinite type interval translation mappings in this class. With respect to this measure, a.e. transformation ...
Added: September 17, 2025
S. V. Bashkevich, A. A. Yelizarov, I. V. Nazarov et al., , in: 2024 Systems of Signal Synchronization, Generating and Processing in Telecommunications (SYNCHROINFO).: IEEE, 2024. P. 1–5.
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Bayramov Ilyas, / Series math "arxiv.org". 2024.
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Boykov I., Boykova A., Potapov A. et al., , in: 14th Chaotic Modeling and Simulation International Conference.: Springer, 2022. Ch. 7 P. 81–95.
The paper consists of three parts. The first one is devoted to approximate methods for evaluating Riemann integrals, singular and hypersingular integrals on closed non-rectifiable curves and fractals in the complex plane. An integral on non-rectifiable curves or fractals is defined as a double integral over a region that bounded by a non-rectifiable curve or ...
Added: January 15, 2023
Gladkov N., Kolesnikov A., Zimin A., Journal of Mathematical Analysis and Applications 2022 Vol. 506 No. 2 Article 125666
The multistochastic Monge–Kantorovich problem on the product X=∏i=1nXi of n spaces is a generalization of the multimarginal Monge–Kantorovich problem. For a given integer number 1≤k<n we consider the minimization problem ∫cdπ→inf on the space of measures with fixed projections onto every Xi1×…×Xik for arbitrary set of k indices {i1,…,ik}⊂{1,…,n}. In this paper we study basic properties of the multistochastic problem, including well-posedness, existence of a dual solution, boundedness and continuity of a dual ...
Added: December 4, 2021
Springer Nature Switzerland AG, 2019.
Gathering the proceedings of the 11th CHAOS2018 International Conference, this book highlights recent developments in nonlinear, dynamical and complex systems. The conference was intended to provide an essential forum for Scientists and Engineers to exchange ideas, methods, and techniques in the field of Nonlinear Dynamics, Chaos, Fractals and their applications in General Science and the ...
Added: October 29, 2021
Korotyaev Evgeny, Saburova N., St Petersburg Mathematical Journal 2019 Vol. 30 P. 667–698
Normalized Laplacians and their perturbations by periodic potentials (Schrödinger operators) on periodic discrete graphs are treated. The spectrum of such an operator consists of an absolutely continuous part, which is the union of a finite number of nondegenerate bands, and a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. Estimates for the Lebesgue ...
Added: February 5, 2021
Korotyaev E., Слоущ В., Алгебра и анализ 2020 Т. 32 № 1 С. 12–39
We consider a periodic Schrödinger operator H on a discrete
periodic graph. Estimates of the discrete spectrum
perturbed operator
decreasing potential. In the case of a potential with a power-law
the asymptotics at infinity is found
asymptotics of the discrete spectrum of the operator for
large coupling constant. ...
Added: February 5, 2021
E. Korotyaev, Russsisan Journal of Math. Physics, Springer 2020 Vol. 27 P. 82–98
We consider 3-dimensonal Schr¨odinger operators with complex potential. We
obtain new trace formulas with new terms, associated with singular measure. In order to
prove these results, we study analytic properties of a modified Fredholm determinant as a
function from Hardy spaces in the upper half-plane. In fact, we reformulate spectral theory
problems as problems of analytic functions from Hardy ...
Added: February 5, 2021
Evgeny Korotyaev, Saburova N., Reviews in Mathematical Physics 2020 Vol. 32 Article 2050024
We consider the Laplacian on a periodic metric graph and obtain its decomposition into a
direct ber integral in terms of the corresponding discrete Laplacian. Eigenfunctions and
eigenvalues of the ber metric Laplacian are expressed explicitly in terms of eigenfunctions
and eigenvalues of the corresponding ber discrete Laplacian and eigenfunctions
of the Dirichlet problem on the unit interval. We ...
Added: February 5, 2021
Chernyshov A., Kozelov B. V., Mogilevsky M. M., Journal of Atmospheric and Solar-Terrestrial Physics 2017 Vol. 161 P. 127–133
In this work, values of the fractal dimension and the connectivity index characterizing the structure of Hall conductivities on the night side of the auroral ionosphere are derived in general form. Restrictions imposed on fractal structure of the ionospheric conductivity are analyzed in terms of the percolation of the ionospheric Hall currents. It is shown ...
Added: November 28, 2019
Trubochkina N. K., Rolich A., В кн.: Запись и воспроизведение объёмных изображений в кинематографе, науке, образовании и в других областях: XI Международная научно-практическая конференция, Москва, 18–19 апреля 2019 г.: Материалы и доклады.: М.: ИПП «КУНА», 2019. Гл. 9 С. 109–124.
The article describes experiments to determine the system of optimal synthesis parameters for fractal and mixed content that meets the requirements of its comfortable viewing in stand-alone helmets of virtual reality. ...
Added: October 30, 2019
Gladkov N., Kolesnikov A., Zimin A., Calculus of Variations and Partial Differential Equations 2019 Vol. 58 No. 173 P. 1–33
The multistochastic (n, k)-Monge–Kantorovich problem on a product space ∏ni=1Xi∏i=1nXi is an extension of the classical Monge–Kantorovich problem. This problem is considered on the space of measures with fixed projections onto Xi1×⋯×XikXi1×⋯×Xik for all k-tuples {i1,…,ik}⊂{1,…,n}{i1,…,ik}⊂{1,…,n} for a given 1≤k<n1≤k<n. In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution ππ to the following important model ...
Added: October 9, 2019
Bendikov A., Grigor'yan A., Molchanov S., / Series arXiv "math". 2018. No. 1811.05210.
Added: November 9, 2018
Skripchenko A., Troubetzkoy S., Journal of London Mathematical Society 2018 Vol. 97 No. 2 P. 149–169
We prove linear upper and lower bounds for the Hausdorff dimension of the set of minimal interval exchange transformations with flips (fIETs); in particular without periodic points, and a linear lower bound for the Hausdorff dimension of the set of non‐uniquely ergodic minimal fIETs. ...
Added: October 3, 2018
Gladkov N., Kolesnikov A., Zimin A., / Series arXiv "math". 2018.
The multistochastic (n,k)-Monge--Kantorovich problem on a product space ∏ni=1Xi is an extension of the classical Monge--Kantorovich problem. This problem is considered on the space of measures with fixed projections onto Xi1×…×Xik for all k-tuples {i1,…,ik}⊂{1,…,n} for a given 1≤k<n. In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution π to the following important model case: n=3,k=2,Xi=[0,1], ...
Added: July 31, 2018