Ing. Tomáš Kalvoda, Ph.D.

Článek

10.1088/1751-8121/ab8b3a

Katedra aplikované matematiky

The Laplace-Beltrami operator in the curved Möbius strip is investigated in the limit when the width of the strip tends to zero. By establishing a norm-resolvent convergence, it is shown that spectral properties of the operator are approximated well by an unconventional flat model whose spectrum can be computed explicitly in terms of Mathieu functions. Contrary to the traditional flat Möbius strip, our effective model contains a geometric potential. A comparison of the three models is made and analytical results are accompanied by numerical computations.

Článek

10.1063/5.0011201

Katedra aplikované matematiky

We study an infinite one-dimensional Ising spin chain where each particle interacts only with its nearest neighbours and is in contact with a heat bath with temperature decaying hyperbolically along the chain. The time evolution of the magnetization (spin expectation value) is governed by a semi-infinite Jacobi matrix. The matrix belongs to a three-parameter family of Jacobi matrices whose spectral problem turns out to be solvable in terms of the basic hypergeometric series. As a consequence, we deduce the essential properties of the corresponding orthogonal polynomials, which seem to be new. Finally, we return to the Ising model and study the time evolution of magnetization and two-spin correlations.

Článek

10.1063/1.5030517

Katedra aplikované matematiky

We study motion of a charged particle confined to a Dirichlet layer of a fixed width placed into a homogeneous magnetic field. If the layer is planar and the field is perpendicular to it, the spectrum consists of infinitely degenerate eigenvalues. We consider translationally invariant geometric perturbations and derive several sufficient conditions under which a magnetic transport is possible, that is, the spectrum, in its entirety or a part of it, becomes absolutely continuous.

Článek

10.1080/03081087.2015.1064348

Katedra aplikované matematiky

A three-parameter family B = B(a, b, c) of weighted Hankel matrices is introduced where a, b, c are positive parameters fulfilling a < b + c, b < a + c, c ≤ a + b. The famous Hilbert matrix is included as a particular case. The direct sum B(a, b, c) ⊕ B(a + 1, b + 1, c) is shown to commute with a discrete analogue 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 ≤ 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.

Článek

Katedra aplikované matematiky

Prezentujeme základní použití otevřeného počítačového algebraického systému SageMath a zamýšlíme se nad významem otevřenosti matematického software.

Článek

10.1016/j.aop.2011.06.007

Katedra teoretické informatiky

We consider a nonrelativistic quantum charged particle moving on a plane under the influence of a uniform magnetic field and driven by a periodically time-dependent Aharonov-Bohm flux. We observe an acceleration effect in the case when the Aharonov-Bohm flux depends on time as a sinusoidal function whose frequency is in resonance with the cyclotron frequency. In particular, the energy of the particle increases linearly for large times. An explicit formula for the acceleration rate is derived with the aid of the quantum averaging method, and then it is checked against a numerical solution and a very good agreement is found.

Článek

10.1137/100809337

Katedra teoretické informatiky

We study the dynamics of a classical nonrelativistic charged particle moving on a punctured plane under the influence of a uniform 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 deal with an approximation for small amplitudes of the flux which is obtained with the aid of averaging methods. 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. 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.