arXiv subjects

E. Strahov

Publications and source records attributed to E. Strahov.

4 recordsLinked to original sources

arXiv2014

Permanental processes from products of complex and quaternionic induced Ginibre ensembles

We consider products of independent random matrices taken from the induced Ginibre ensemble with complex or quaternion elements. The joint densities for the complex eigenvalues of the product matrix can be written down exactly for a product of any fixed number of matrices and any finite matrix size. We show that the squared absolute values of the eigenvalues form a permanental process, generalising the results of Kostlan and Rider for single matrices to products of complex and quaternionic matrices. Based on these findings, we can first write down exact results and asymptotic expansions for the so-called hole probabilities, that a disc centered at the origin is void of eigenvalues. Second, we compute the asymptotic expansion for the opposite problem, that a large fraction of complex eigenvalues occupies a disc of fixed radius centered at the origin; this is known as the overcrowding problem. While the expressions for finite matrix size depend on the parameters of the induced ensembles, the asymptotic results agree to leading order with previous results for products of square Ginibre matrices.

math-ph↗

arXiv2004

Averages of Characteristic Polynomials in Random Matrix Theory

We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the bulk scaling asymptotic limits are found for ensembles with Gaussian weights. Classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulas by a simple computation. We employ a discrete approximation method: the problem is solved for discrete analogues of random matrix ensembles originating from representation theory, and then a limit transition is performed. Exact pfaffian/determinantal formulas for the discrete averages are proved using standard tools of linear algebra; no application of orthogonal or skew-orthogonal polynomials is needed.

math-ph↗

arXiv2000

Berry's phase for compact Lie groups

The Lie group adiabatic evolution determined by a Lie algebra parameter dependent Hamiltonian is considered. It is demonstrated that in the case when the parameter space of the Hamiltonian is a homogeneous Kähler manifold its fundamental Kähler potentials completely determine Berry geometrical phase factor. Explicit expressions for Berry vector potentials (Berry connections) and Berry curvatures are obtained using the complex parametrization of the Hamiltonian parameter space. A general approach is exemplified by the Lie algebra Hamiltonians corresponding to SU(2) and SU(3) evolution groups.

quant-ph↗

arXiv2000

A geometrical approach to non-adiabatic transitions in quantum theory: applications to NMR, over-barrier reflection and parametric excitation of quantum oscillator

This article deals with non-adiabatic processes (i.e. processes excluded by the adiabatic theorem) from the geometrical (group-theoretical) point of view. An approximated formula for the probabilities of the non-adiabatic transitions is derived in the adiabatic regime for the case when the parameter-dependent Hamiltonian represents a smooth curve in the Lie algebra and the quantal dynamics is determined by the corresponding Lie group evolution operator. We treat the spin precession in a time-dependent magnetic field and the over-barrier reflection problem in a uniform way using the first-order dynamical equations on SU(2) and $SU(1.1)$ group manifolds correspondingly. A comparison with analytic solutions for simple solvable models is provided.

quant-ph↗