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Pavel Stovicek

Publications and source records attributed to Pavel Stovicek.

14 recordsLinked to original sources
arXiv2022

A family of orthogonal polynomials corresponding to Jacobi matrices with a trace class inverse

Assume that $\{a_{n};\,n\geq0\}$ is a sequence of positive numbers and $\sum a_{n}^{\,-1}<\infty$. Let $α_{n}=ka_{n}$, $β_{n}=a_{n}+k^{2}a_{n-1}$ where $k\in(0,1)$ is a parameter, and let $\{P_{n}(x)\}$ be an orthonormal polynomial sequence defined by the three-term recurrence \[ α_{0}P_{1}(x)+(β_{0}-x)P_{0}(x)=0,\ α_{n}P_{n+1}(x)+(β_{n}-x)P_{n}(x)+α_{n-1}P_{n-1}(x)=0 \] for $n\geq1$, with $P_{0}(x)=1$. Let $J$ be the corresponding Jacobi (tridiagonal) matrix, i.e. $J_{n,n}=β_{n}$, $J_{n,n+1}=J_{n+1,n}=α_{n}$ for $n\geq0$. Then $J^{-1}$ exists and belongs to the trace class. We derive an explicit formula for $P_{n}(x)$ as well as for the characteristic function of $J$ and describe the orthogonality measure for the polynomial sequence. As a particular case, the modified $q$-Laguerre polynomials are introduced and studied.

math-ph↗
arXiv2019

On infinite Jacobi matrices with a trace class resolvent

Let $\{\hat{P}_{n}(x)\}$ be an orthonormal polynomial sequence and denote by $\{w_{n}(x)\}$ the respective sequence of functions of the second kind. Suppose the Hamburger moment problem for $\{\hat{P}_{n}(x)\}$ is determinate and denote by $J$ the corresponding Jacobi matrix operator on $\ell^{2}$. We show that if $J$ is positive definite and $J^{-1}$ belongs to the trace class then the series on the right-hand side of the defining equation \[ \mathfrak{F}(z):=1-z\sum_{n=0}^{\infty}w_{n}(0)\hat{P}_{n}(z) \] converges locally uniformly on $\mathbb{C}$ and it holds true that $\mathfrak{F}(z)=\prod_{n=1}^{\infty}(1-z/λ_{n})$ where $\{λ_{n};\,n=1,2,3,\ldots\}=\mathrm{Spec}\,J$. Furthermore, the Al-Salam-Carlitz II polynomials are treated as an example of orthogonal polynomials to which this theorem can be applied.

math.SP↗
arXiv2016

The heat kernel for two Aharonov-Bohm solenoids in a uniform magnetic field

A non-relativistic quantum model is considered with a point particle carrying a charge $e$ and moving on the plane pierced by two infinitesimally thin Aharonov-Bohm solenoids and subjected to a perpendicular uniform magnetic field of magnitude $B$. Relying on a technique due to Schulman and Sunada which is applicable to Schrödinger operators on multiply connected configuration manifolds a formula is derived for the corresponding heat kernel. As an application of the heat kernel formula, an approximate asymptotic expressions are derived for the lowest eigenvalue lying above the first Landau level and for the corresponding eigenfunction while assuming that $|eB|R^{2}/(\hbar c)$ is large where $R$ is the distance between the two solenoids.

math-ph↗
arXiv2015

Family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

A three-parameter family $B=B(a,b,c)$ of weighted Hankel matrices is introduced with the entries \[ B_{j,k}=\frac{Γ(j+k+a)}{Γ(j+k+b+c)}\,\sqrt{\frac{Γ(j+b)Γ(j+c)Γ(k+b)Γ(k+c)}{Γ(j+a)\, j!\,Γ(k+a)\, k!}}\,, \] $j,k\in\mathbb{Z}_{+}$, supposing $a$, $b$, $c$ are positive and $a<b+c$, $b<a+c$, $c\leq a+b$. The famous Hilbert matrix is included as a particular case. The direct sum $B(a,b,c)\oplus B(a+1,b+1,c)$ is shown to commute with a discrete analog of the dilatation operator. It follows that there exists a three-parameter family of real symmetric Jacobi matrices, $T(a,b,c)$, commuting with $B(a,b,c)$. The orthogonal polynomials associated with $T(a,b,c)$ turn out to be the continuous dual Hahn polynomials. Consequently, a unitary mapping $U$ diagonalizing $T(a,b,c)$ can be constructed explicitly. At the same time, $U$ diagonalizes $B(a,b,c)$ and the spectrum of this matrix operator is shown to be purely absolutely continuous and filling the interval $[0,M(a,b,c)]$ where $M(a,b,c)$ is known explicitly. If the assumption $c\leq a+b$ is relaxed while the remaining inequalities on $a$, $b$, $c$ are all supposed to be valid, the spectrum contains also a finite discrete part lying above the threshold $M(a,b,c)$. Again, all eigenvalues and eigenvectors are described explicitly.

math.SP↗
arXiv2014

The Hahn-Exton $q$-Bessel function as the characteristic function of a Jacobi matrix

A family $\mathcal{T}^{(ν)}$, $ν\in\mathbb{R}$, of semiinfinite positive Jacobi matrices is introduced with matrix entries taken from the Hahn-Exton $q$-difference equation. The corresponding matrix operators defined on the linear hull of the canonical basis in $\ell^{2}(\mathbb{Z}_{+})$ are essentially self-adjoint for $|ν|\geq1$ and have deficiency indices $(1,1)$ for $|ν|<1$. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzed in detail. The spectrum is discrete and the characteristic equation on eigenvalues is derived explicitly in all cases. Particularly, the Hahn-Exton $q$-Bessel function $J_ν(z;q)$ serves as the characteristic function of the Friedrichs extension. As a direct application one can reproduce, in an alternative way, some basic results about the $q$-Bessel function due to Koelink and Swarttouw.

math.SP↗
arXiv2014

Orthogonal polynomials associated with Coulomb wave functions

A class of orthogonal polynomials associated with Coulomb wave functions is introduced. These polynomials play a role analogous to that the Lommel polynomials do in the theory of Bessel functions. The measure of orthogonality for this new class is described explicitly. In addition, the orthogonality measure problem is also discussed on a more general level. Apart of this, various identities derived for the new orthogonal polynomials may be viewed as generalizations of some formulas known from the theory of Bessel functions. A key role in these derivations is played by a Jacobi (tridiagonal) matrix $J_{L}$ whose eigenvalues coincide with reciprocal values of the zeros of the regular Coulomb wave function $F_{L}(η,ρ)$. The spectral zeta function corresponding to the regular Coulomb wave function or, more precisely, to the respective tridiagonal matrix is studied as well.

math.CA↗
arXiv2013

Special functions and spectrum of Jacobi matrices

Several examples of Jacobi matrices with an explicitly solvable spectral problem are worked out in detail. In all discussed cases the spectrum is discrete and coincides with the set of zeros of a special function. Moreover, the components of corresponding eigenvectors are expressible in terms of special functions as well. Our approach is based on a recently developed formalism providing us with explicit expressions for the characteristic function and eigenvectors of Jacobi matrices. This is done under an assumption of a simple convergence condition on matrix entries. Among the treated special functions there are regular Coulomb wave functions, confluent hypergeometric functions, q-Bessel functions and q-confluent hypergeometric functions. In addition, in the case of q-Bessel functions, we derive several useful identities.

math.SP↗
arXiv2011

Resonant cyclotron acceleration of particles by a time periodic singular flux tube

We study the dynamics of a classical nonrelativistic charged particle moving on a punctured plane under the influence of a homogeneous magnetic field and driven by a periodically time-dependent singular flux tube through the hole. We observe an effect of resonance of the flux and cyclotron frequencies. The particle is accelerated to arbitrarily high energies even by a flux of small field strength which is not necessarily encircled by the cyclotron orbit; the cyclotron orbits blow up and the particle oscillates between the hole and infinity. We support this observation by an analytic study of an approximation for small amplitudes of the flux which is obtained with the aid of averaging methods. This way we derive asymptotic formulas that are afterwards shown to represent a good description of the accelerated motion even for fluxes which are not necessarily small. More precisely, we argue that the leading asymptotic terms may be regarded as approximate solutions of the original system in the asymptotic domain as the time tends to infinity.

math-ph↗
arXiv2010

Noncommutative Bloch analysis of Bochner Laplacians with nonvanishing gauge fields

Given an invariant gauge potential and a periodic scalar potential \tilde{V} on a Riemannian manifold \tilde{M} with a discrete symmetry group Γ, consider a Γ-periodic quantum Hamiltonian \tilde{H}=-\tildeΔ_{B}+\tilde{V} where \tildeΔ_{B} is the Bochner Laplacian. Both the gauge group and the symmetry group Γcan be noncommutative, and the gauge field need not vanish. On the other hand, Γis supposed to be of type I. To any unitary representation Λof Γone relates a Hamiltonian H^Λ=-Δ_{B}^Λ+V on M=\tilde{M}/Γwhere V is the projection of \tilde{V} to M. We describe a construction of the Bloch decomposition of \tilde{H} into a direct integral whose components are H^Λ, with Λrunning over the dual space \hatΓ. The evolution operator and the resolvent decompose correspondingly. Conversely, given Λ\in\hatΓ, one can express the propagator \mathcal{K}_{t}^Λ(y_{1},y_{2}) (the kernel of \exp(-itH^Λ)) in terms of the propagator \tilde{\mathcal{K}}_{t}(y_{1},y_{2}) (the kernel of \exp(-it\tilde{H})) as a weighted sum over Γ. Such a formula is known in theoretical physics for the case when the gauge field vanishes and \tilde{M} is a universal covering of a multiply connected space M. We show that these constructions are mutually inverse. Analogous formulas exist for resolvents and their kernels (Green functions) as well.

math-ph↗
arXiv2007

On the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum

We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E_n~n^α, with 0<α<1. In particular, the gaps between successive eigenvalues decay as n^{α-1}. V(t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with respect to the spectral decomposition of H obey the estimate |V(t)_{m,n}|<=ε*|m-n|^{-p}max{m,n}^{-2γ} for m!=n where ε>0, p>=1 and γ=(1-α)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and εis small enough. More precisely, for any initial condition Ψ\in Dom(H^{1/2}), the diffusion of energy is bounded from above as _Ψ(t)=O(t^σ) where σ=α/(2\ceil{p-1}γ-1/2). As an application we consider the Hamiltonian H(t)=|p|^α+ε*v(θ,t) on L^2(S^1,dθ) which was discussed earlier in the literature by Howland.

math-ph↗
arXiv2007

On the stability of periodically time-dependent quantum systems

The main motivation of this article is to derive sufficient conditions for dynamical stability of periodically driven quantum systems described by a Hamiltonian H(t), i.e., conditions under which it holds sup_{t in R} | (psi(t),H(t) psi(t)) |<\infty where psi(t) denotes a trajectory at time t of the quantum system under consideration. We start from an analysis of the domain of the quasi-energy operator. Next we show, under certain assumptions, that if the spectrum of the monodromy operator U(T,0) is pure point then there exists a dense subspace of initial conditions for which the mean value of energy is uniformly bounded in the course of time. Further we show that if the propagator admits a differentiable Floquet decomposition then || H(t) psi(t) || is bounded in time for any initial condition psi(0), and one employs the quantum KAM algorithm to prove the existence of this type of decomposition for a fairly large class of H(t). In addition, we derive bounds uniform in time on transition probabilities between different energy levels, and we also propose an extension of this approach to the case of a higher order of differentiability of the Floquet decomposition. The procedure is demonstrated on a solvable example of the periodically time-dependent harmonic oscillator.

math-ph↗
arXiv2001

Construction of raising and lowering operators for Uq(sl(2,R))

In the case of Uq(sl(2,R)) at root of unity q-deformed analogues are proposed for the generator of the maximal compact subalgebra, J, and for the raising and lowering operators. We prove an algebraic identity which implies that J has similar properties as in the nondeformed case.

math.QA↗
arXiv2001

Generalised boundary conditions for the Aharonov-Bohm effect combined with a homogeneous magnetic field

The most general admissible boundary conditions are derived for an idealised Aharonov-Bohm flux intersecting the plane at the origin on the background of a homogeneous magnetic field. A standard technique based on self-adjoint extensions yields a four-parameter family of boundary conditions; other two parameters of the model are the Aharonov-Bohm flux and the homogeneous magnetic field. The generalised boundary conditions may be regarded as a combination of the Aharonov-Bohm effect with a point interaction. Spectral properties of the derived Hamiltonians are studied in detail.

quant-ph↗