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Dirac fermion in a time-dependent spherical box
(České vysoké učení technické v Praze, 2025) Matchonov, Kuvonchbek; Matrasulov, Davron; Dittrich, Jaroslav
We consider a Dirac particle in a spherical box with a time-dependent radius. Analytical and numerical solutions of the time-dependent Dirac equation with time-dependent boundary (Dirichlet) conditions are obtained. Using the obtained solutions, physically observable characteristics of the dynamical confinement, such as the average kinetic energy (as a function of time) and average quantum force acting on the particle by the moving wall, are calculated. The trembling motion is analysed by computing the average coordinate of the Dirac particle as a function of time. The absence of the geometric phase is shown by direct calculation.
Method of lines for reaction-diffusion systems admitting invariant regions
(České vysoké učení technické v Praze, 2025) van der Meer, Niels; Beneš, Michal
Systems of nonlinear reaction-diffusion equations arise in various fields, including chemistry, population dynamics, pattern formation, phase transitions, and image processing. With the exception of few analytically solvable cases, they are treated by numerical methods carefully adjusted to capture the nonlinear phenomena exhibited by the solution. This article shows how to extend the notion of invariant regions generalizing the maximum principle for diffusion equations to the finite-difference method of lines, and how to consequently prove convergence of the underlying numerical scheme. We also provide two particular examples of reaction-diffusion systems in one-dimensional space with a diagonal diffusion operator, which are solved by the presented numerical method.
Classical and quantum superintegrable systems on the sphere and the hyperbolic 2-space
(České vysoké učení technické v Praze, 2025) del Olmo, Mariano A.; Romaniega, Álvaro
We present two superintegrable Hamiltonian systems in two dimensions, defined on the sphere and on the hyperbolic plane. These systems are generalised à la Tremblay-Turbiner-Winternitz (TTW), involving the introduction of a real parameter k > 0, with the aim of extending superintegrable Hamiltonian systems to curved spaces in a way similar to the TTW system on the plane. We carry out both classical and quantum analyses of these new systems. We prove that the superintegrability of the initial systems (i.e. when k = 1) is preserved when k is rational, as in the TTW case. A detailed study of their classical counterparts and trajectories is also included.
Unitary generation of GHZ states
(České vysoké učení technické v Praze, 2025) Bandilla, Arno; Chadzitaskos, Goce; Jex, Igor
Entangled states represent fascinating elements of quantum mechanics and quantum technology. Their generation is an interesting problem combining many techniques developed in quantum optics. Using algebraic properties of special transforms we derive the explicit form of the effective Hamiltonian that enables the generation of highly nonclassical states of the GHZ type. Such Hamiltonians belong to the higher order Hamiltonians often discussed in nonlinear quantum optics. We derive the form of the effective Hamiltonian that enables the generation of highly nonclassical states of the GHZ type.
gl3 ALGEBRA IN MIXED MATRIX REPRESENTATIONS
(České vysoké učení technické v Praze, 2025) Turbiner, Alexander V.
It is demonstrated that the so-called mixed realization of the gl3 algebra generators in terms of matrix differential operators in two variables, as presented by Smirnov-Turbiner (2013), can be “lifted” into the action in the Fock space associated with the five-dimensional Heisenberg algebra h5. A realization of the gl3 generators in terms of matrix finite-difference (translation-invariant) operators, matrix discrete (dilatation-invariant) operators, matrix complex operators in (z, z̄), and their mixtures is presented.