On periodic $p$-harmonic functions on Cayley tree

On Ground States of Rozikov Model on the Cayley Tree

In this paper we consider a model on a Cayley tree which has a finite radius of interactions, the model was first considered by Rozikov. We describe a set of periodic ground states of the model.

Shorting, parallel addition and form sums of nonnegative selfadjoint linear relations

We extend the operations of shorting and parallel addition from the cone of bounded nonnegative selfadjoint operators in a Hilbert space to the set of all nonnegative selfadjoint linear relations. New properties of these operations and connections with the Cayley transforms are established. It is shown that for a pair of nonnegative selfadjoint linear relations there exist, in general, two arithmetic--harmonic means. Applications of the arithmetic, harmonic, and arithmetic--harmonic means to the theory of nonnegative selfadjoint extensions of nonnegative symmetric linear relations are given.

Harmonic analysis on Cayley Trees II: the Bose Einstein condensation

We investigate the Bose-Einstein Condensation on non homogeneous non amenable networks for the model describing arrays of Josephson junctions on perturbed Cayley Trees. The resulting topological model has also a mathematical interest in itself. The present paper is then the application to the Bose-Einstein Condensation phenomena, of the harmonic analysis aspects arising from additive and density zero perturbations, previously investigated by the author in a separate work. Concerning the appearance of the Bose-Einstein Condensation, the results are surprisingly in accordance with the previous o…

Periodic points of a $p$-adic operator and their $p$-adic Gibbs measures

In this paper we investigate generalized Gibbs measure (GGM) for $p$-adic Hard-Core(HC) model with a countable set of spin values on a Cayley tree of order $k\geq 2$. This model is defined by $p$-adic parameters $λ_i$, $i\in \mathbb N$. We analyze $p$-adic functional equation which provides the consistency condition for the finite-dimensional generalized Gibbs distributions. Each solutions of the functional equation defines a GGM by $p$-adic version of Kolmogorov's theorem. We define $p$-adic Gibbs distributions as limit of the consistent family of finite-dimensional generalized Gibbs distribu…

On quasi-transitive amenable graphs

The existence of nonconstant harmonic Dirichlet functions on a Cayley graph of a discrete group is equivalent to the nonvanishing of the first L2-cohomology of the given group. It was first proven by Cheeger and Gromov that such functions do not exists on the Cayley-graph of an amenable group. The result was extended using Foster's averaging formula to transitive amenable graphs by Medolla and Soardi. In this paper we extend this result further for amenable graphs which are transitive in the category of quasi-isometries. Such graphs are usually not quasi-isometric to transitive graphs. The ext…

带拟周期有界势的一维量子谐振子的可约性研究

On the reducibility of the 1d quantum harmonic oscillator with a quasi-periodic bounded potential

该研究利用厄米基下微扰矩阵的对角线衰减性，通过KAM迭代证明了带仅有界、空间导数至多按$|x|^δ(δ<1)$增长的拟周期势的一维量子谐振子在$L^2(\mathbb{R})$中的可约性。