Harmonic analysis on perturbed Cayley Trees

Fidaleo, Francesco

doi:10.1016/j.jfa.2011.04.007

Cited by 10 publications

(58 citation statements)

References 12 publications

(52 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Denote P n the orthogonal projection in B( 2 (V X) associated to the finite region V n . We report the definition of the the integrated density of the states of a bounded self-adjoint operator B ∈ B( 2 (V X)) given in [6]. Indeed, consider on B( 2 (V X)) the state τ n := 1 |V n | Tr n (P n · P n ).…”

Section: Definition 21mentioning

confidence: 99%

“…When the graph is amenable and the operator B has finite propagation, the definition and some of the main facts relative to the IDS considerably simplify as the boundary effects play no role in the infinite volume limit, see Theorem 2.1 of [6].…”

Section: Definition 21mentioning

confidence: 99%

“…The following theorem collects Proposition 1.3 of [6] and Corollary 2.6 of [5], which we report for the convenience of the reader.…”

Section: Definition 21mentioning

confidence: 99%

“…6 The following theorem collects the implications of the appearance of Hidden Spectrum, relatively to the critical density.…”

Section: The Dynamics Generated By the Bogoliubov Transformationsmentioning

confidence: 99%

“…The investigation of the BEC for free Bosons on homogeneous Cayley Trees has been started in the seminal paper [19]. The surprising fact is that, even for additive perturbations of Cayley Trees, that is in non amenable situations for which the boundary effects cannot be neglected in the infinite volume limit, most of the previous results still hold true, see [5,7]. The present paper is mainly devoted to prove systematically the relevant spectral results relative to the Adjacency listed above, and then to investigate in the full generality the thermodynamics of the Pure Hopping models for the networks listed in the table above, relatively to the non trivial cases N, N Z d , and Z d Z.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Harmonic Analysis on Inhomogeneous Amenable Networks and the Bose–Einstein Condensation

Fidaleo

2015

J Stat Phys

Self Cite

View full text Add to dashboard Cite

We study in detail relevant spectral properties of the adjacency matrix of inhomogeneous amenable networks, and in particular those arising by negligible additive perturbations of periodic lattices. The obtained results are deeply connected to the systematic investigation of the Bose-Einstein condensation for the so called Pure Hopping model describing the thermodynamics of Cooper pairs in arrays of Josephson junctions. After a careful investigation of the infinite volume limits of the finite volume adjacency matrix corresponding to the (opposite of the) Hamiltonian of the system, the main results can be summarised as follows. First, the appearance of the Hidden Spectrum for the Integrated Density of the States in the region close to the bottom of the Hamiltonian, implies that the critical density is always finite. Second, we show that the Bose-Einstein condensation can appear if and only if the adjacency matrix is transient, and not just when the critical density is finite. We can then exhibit examples of networks for which condensation effects can appear in a natural way, even if the critical density is infinite and vice-versa, that is when the critical density is finite but the system does not admit any locally normal state exhibiting condensation. Contrarily to the known homogeneous examples, we also exhibit networks whose geometrical dimension is less than 3, for which the condensation takes place. Due to inhomogeneity, the spatial distribution of the condensate described by the shape of the ground state wave-function (i.e. the Perron-Frobenius weight), is non homogeneous as well: the particles condensate even in the configuration space. Such a spatial distribution of the condensate is connected with the Perron-Frobenius dimension defined in a natural way. For systems for which the critical density is finite and the adjacency matrix is transient, we show that, if the Perron-Frobenius dimension is greater that the geometrical one, we can have condensation only if the mean density of the state is infinite. Conversely, in the opposite situation when the geometrical dimension exceeds the Perron-Frobenius one, the condensation appears only for states with mean density coinciding A Berta con profonda stima e gratitudine. B Francesco Fidaleo

show abstract

Section: Definition 21mentioning

confidence: 99%

Section: Definition 21mentioning

confidence: 99%

“…The following theorem collects Proposition 1.3 of [6] and Corollary 2.6 of [5], which we report for the convenience of the reader.…”

Section: Definition 21mentioning

confidence: 99%

“…6 The following theorem collects the implications of the appearance of Hidden Spectrum, relatively to the critical density.…”

Section: The Dynamics Generated By the Bogoliubov Transformationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Harmonic Analysis on Inhomogeneous Amenable Networks and the Bose–Einstein Condensation

Fidaleo

2015

J Stat Phys

Self Cite

View full text Add to dashboard Cite

show abstract

Corrigendum to “Harmonic analysis on perturbed Cayley Trees” [J. Funct. Anal. 261 (3) (2011) 604–634]

Fidaleo

2012

Journal of Functional Analysis

View full text Add to dashboard Cite

A proposal for the thermodynamics of certain open systems

Fidaleo

Viaggiu

2017

Physica A: Statistical Mechanics and its Applications

Self Cite

View full text Add to dashboard Cite

Motivated by the fact that the (inverse) temperature might be a function of the energy levels in the Planck distribution $n_\epsilon=\frac1{\zeta^{-1}e^{\beta(\epsilon)\epsilon}-1}$ for the occupation number $n_\epsilon$ of the level $\epsilon$, we show that it can be naturally achieved by imposing the constraint concerning the conservation of a weighted sum $\sum_{\epsilon}f(\epsilon)\epsilon n_\epsilon$, with a fixed positive weight function $f$, of the contributions of the single energy levels occupation in the Microcanonical Ensemble scheme, obtaining $\beta(\epsilon)\propto f(\epsilon)$. This immediately addresses the possibility that also a weighted sum $\sum_{\epsilon}g(\epsilon)n_\epsilon$ of the particles occupation number is conserved, having as a consequence that the chemical potential might be a function of the energy levels of the system as well. This scheme leads to a thermodynamics of open systems in the following way: the equilibrium is reached when the entropy function is maximised under the constraints that some weighed sums of occupation of the energy levels and the occupation numbers are conserved. The standard case of isolated systems corresponds to the weight functions being trivial (i.e. $f, g$ are identically 1). For such open systems, new and unexpected phenomena which might happen in nature can appear, like the Bose Einstein Condensation in excited levels. The ideas outlined in the present paper may provide a new approach for the treatment of the irreversible thermodynamics.Comment: 20 pages, to appear in Physica

show abstract

Harmonic analysis on perturbed Cayley Trees

Cited by 10 publications

References 12 publications

Harmonic Analysis on Inhomogeneous Amenable Networks and the Bose–Einstein Condensation

Harmonic Analysis on Inhomogeneous Amenable Networks and the Bose–Einstein Condensation

Corrigendum to “Harmonic analysis on perturbed Cayley Trees” [J. Funct. Anal. 261 (3) (2011) 604–634]

A proposal for the thermodynamics of certain open systems

Contact Info

Product

Resources

About