Related papers: Eigenvalues, absolute continuity and localizations…
First we report that the adjacency matrices of real-world complex networks systematically have null eigenspaces with much higher dimensions than that of random networks. These null eigenvalues are caused by duplication mechanisms leading to…
It is well established that the physical phenomenon of intermittency can be investigated via the spectral analysis of a transfer operator associated with the dynamics of an interval map with indifferent fixed point. We present here for the…
We study the localization transitions which arise in both one and two dimensions when quantum mechanical particles described by a random Schr\"odinger equation are subjected to a constant imaginary vector potential. A path-integral…
It is known that in quantum field theory, localized operations, e.g.\ given by unitary operators in local observable algebras, may lead to non-causal, or superluminal, state changes within their localization region. In this article, it is…
Within the framework of quantum harmonic analysis, for a locally compact group $G$ with a square-integrable, irreducible unitary representation, we analyze the eigenvalue distributions of convolutions between indicator functions on $G$ and…
We consider some random band matrices with band-width $N^\mu$ whose entries are independent random variables with distribution tail in $x^{-\alpha}$. We consider the largest eigenvalues and the associated eigenvectors and prove the…
We study the value distribution and extreme values of eigenfunctions for the ``quantized cat map''. This is the quantization of a hyperbolic linear map of the torus. In a previous paper it was observed that there are quantum symmetries of…
Transition probabilities are an important and useful tool in quantum mechanics. However, in their present form, they are limited in scope and only apply to pure quantum states. In this article we extend their applicability to mixed states…
We study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of $S^+(U^3)$, a matrix based on the amplitudes of walks in the quantum walk,…
A model Hamiltonian is proposed in order to understand the localization-delocalization transition in a quantum dot, where there are two gate voltages: top and side. Considering energetically favorable degrees of freedom only, we achieve a…
By using a real matrix translation, we propose a coupled eigenvalue problem for octonionic operators. In view of possible applications in quantum mechanics, we also discuss the hermiticity of such operators. Previous difficulties in…
A continuous-time quantum random walk describes the motion of a quantum mechanical particle on an underlying graph. The graph itself is associated with a Hilbert space of dimension equal to the number of vertices. The dynamics of the walk…
This paper is concerned with generalizations of the notion of principal eigenvalue in the context of space-time periodic cooperative systems. When the spatial domain is the whole space, the Krein-Rutman theorem cannot be applied and this…
We study a class of symmetric quantum walks on Hamming graphs, where the distance between vertices specifies the transition probability. A special model is the simple quantum walk on the hypercube, which has been discussed in the…
We analyze the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations. Although, in general, these quantum operations cannot be diagonalized it is shown that their resulting asymptotic…
A universal energy eigenvalue equation is proposed in this paper. It is proven that the unique set of eigenfunctions or preferred basis exists for any non-isolated sub-system. Applying the new eigenvalue equation to the relative motion of a…
Mathematical analysis of the spectral properties of the time evolution operator in quantum walks is essential for understanding key dynamical behaviors such as localization and long-term evolution. The inhomogeneous three-state case, in…
The eigenvalue spacing of a uniformly chosen random finite unipotent matrix in its permutation action on lines is studied. We obtain bounds for the mean number of eigenvalues lying in a fixed arc of the unit circle and offer an approach…
We propose the assumption of quantum mechanics on a discrete space and time, which implies the modification of mathematical expressions for some postulates of quantum mechanics. In particular we have a Hilbert space where the vectors are…
In this paper we investigate homogenization results for the principal eigenvalue problem associated to $1$-homogeneous, uniformly elliptic, second-order operators. Under rather general assumptions, we prove that the principal eigenpair…