Search results for "Values"
showing 10 items of 1365 documents
Tridiagonality, supersymmetry and non self-adjoint Hamiltonians
2019
In this paper we consider some aspects of tridiagonal, non self-adjoint, Hamiltonians and of their supersymmetric counterparts. In particular, the problem of factorization is discussed, and it is shown how the analysis of the eigenstates of these Hamiltonians produce interesting recursion formulas giving rise to biorthogonal families of vectors. Some examples are proposed, and a connection with bi-squeezed states is analyzed.
Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices
2017
For $$k,m,n\in {\mathbb {N}}$$ , we consider $$n^k\times n^k$$ random matrices of the form $$\begin{aligned} {\mathcal {M}}_{n,m,k}({\mathbf {y}})=\sum _{\alpha =1}^m\tau _\alpha {Y_\alpha }Y_\alpha ^T,\quad {Y}_\alpha ={\mathbf {y}}_\alpha ^{(1)}\otimes \cdots \otimes {\mathbf {y}}_\alpha ^{(k)}, \end{aligned}$$ where $$\tau _{\alpha }$$ , $$\alpha \in [m]$$ , are real numbers and $${\mathbf {y}}_\alpha ^{(j)}$$ , $$\alpha \in [m]$$ , $$j\in [k]$$ , are i.i.d. copies of a normalized isotropic random vector $${\mathbf {y}}\in {\mathbb {R}}^n$$ . For every fixed $$k\ge 1$$ , if the Normalized Counting Measures of $$\{\tau _{\alpha }\}_{\alpha }$$ converge weakly as $$m,n\rightarrow \infty $$…
Asymptotics for pooled marginal slicing estimator based on SIRα approach
2005
Pooled marginal slicing (PMS) is a semiparametric method, based on sliced inverse regression (SIR) approach, for achieving dimension reduction in regression problems when the outcome variable y and the regressor x are both assumed to be multidimensional. In this paper, we consider the SIR"@a version (combining the SIR-I and SIR-II approaches) of the PMS estimator and we establish the asymptotic distribution of the estimated matrix of interest. Then the asymptotic normality of the eigenprojector on the estimated effective dimension reduction (e.d.r.) space is derived as well as the asymptotic distributions of each estimated e.d.r. direction and its corresponding eigenvalue.
Optimal designs for a one-way layout with covariates
2000
Abstract For the general class of Φ q -criteria optimal designs are characterized which reflect the inherent symmetry in a one-way layout with covariates. In particular, the eigenvalues of the covariance matrices are related to those in suitably chosen marginal models depending on the underlying interaction structure.
A non-linear optimization procedure to estimate distances and instantaneous substitution rate matrices under the GTR model.
2006
Abstract Motivation: The general-time-reversible (GTR) model is one of the most popular models of nucleotide substitution because it constitutes a good trade-off between mathematical tractability and biological reality. However, when it is applied for inferring evolutionary distances and/or instantaneous rate matrices, the GTR model seems more prone to inapplicability than more restrictive time-reversible models. Although it has been previously noted that the causes for intractability are caused by the impossibility of computing the logarithm of a matrix characterised by negative eigenvalues, the issue has not been investigated further. Results: Here, we formally characterize the mathematic…
The Concept of Duality and Applications to Markov Processes Arising in Neutral Population Genetics Models
1999
One possible and widely used definition of the duality of Markov processes employs functions H relating one process to another in a certain way. For given processes X and Y the space U of all such functions H, called the duality space of X and Y, is studied in this paper. The algebraic structure of U is closely related to the eigenvalues and eigenvectors of the transition matrices of X and Y. Often as for example in physics (interacting particle systems) and in biology (population genetics models) dual processes arise naturally by looking forwards and backwards in time. In particular, time-reversible Markov processes are self-dual. In this paper, results on the duality space are presented f…
Complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms
2013
We use the quantum separation of variable (SOV) method to construct the eigenstates of the open XXZ chain with the most general boundary terms. The eigenstates in the inhomogeneous case are constructed in terms of solutions of a system of quadratic equations. This SOV representation permits us to compute scalar products and can be used to calculate form factors and correlation functions.
Some results on the rotated infinitely deep potential and its coherent states
2021
The Swanson model is an exactly solvable model in quantum mechanics with a manifestly non self-adjoint Hamiltonian whose eigenvalues are all real. Its eigenvectors can be deduced easily, by means of suitable ladder operators. This is because the Swanson Hamiltonian is deeply connected with that of a standard quantum Harmonic oscillator, after a suitable rotation in configuration space is performed. In this paper we consider a rotated version of a different quantum system, the infinitely deep potential, and we consider some of the consequences of this rotation. In particular, we show that differences arise with respect to the Swanson model, mainly because of the technical need of working, he…
Duality of reduced density matrices and their eigenvalues
2014
For states of quantum systems of N particles with harmonic interactions we prove that each reduced density matrix ρ obeys a duality condition. This condition implies duality relations for the eigenvalues λk of ρ and relates a harmonic model with length scales ${{\ell }_{1}},{{\ell }_{2}},\ldots ,{{\ell }_{N}}$ with another one with inverse lengths $1/{{\ell }_{1}},1/{{\ell }_{2}},\ldots ,1/{{\ell }_{N}}$. Entanglement entropies and correlation functions inherit duality from ρ. Self-duality can only occur for noninteracting particles in an isotropic harmonic trap.
Non-self-adjoint Hamiltonians with complex eigenvalues
2016
Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.