P. Šťovíček

Quantum mechanics in a discrete space-time

Abstract A complete description of quantum kinematics in the sense of Mackey and Weyl is presented for the class of systems whose underlying configuration spaces are finite sets equipped with the structure of finite Abelian groups. For a given finite Abelian group there is a unique class of unitarily equivalent, irreducible imprimitivity systems in a finite-dimensional Hilbert space. Schwingers tensor product decomposition is extended to this class of systems. The finite analogue of the Galilei group over the finite space-time lattice yields a discrete time evolution operator which is proposed to be the free Hamiltonian.

Floquet Hamiltonians with pure point spectrum

AbstractWe consider Floquet Hamiltonians of the type

Quantum dressing orbits on compact groups

K_F : = - i\partial _t + H_0 + \beta V(\omega t)

QUANTIZATION OF KINEMATICS ON CONFIGURATION MANIFOLDS

, whereH0, a selfadjoint operator acting in a Hilbert space ℋ, has simple discrete spectrumE10 for a given α>0,t↦V(t) is 2π-periodic andr times strongly continuously differentiable as a bounded operator on ℋ, ω and β are real parameters and the periodic boundary condition is imposed in time. We show, roughly, that providedr is large enough, β small enough and ω non-resonant, then the spectrum ofKf is pure point. The method we use relies on a successive application of the adiabatic treatment due to Howland and the KAM-type iteration settled by Bellissard and extended by Combescure. Both tools are revisited, adjusted and at some points slightly simplified.

On the two-dimensional Coulomb-like potential with a central point interaction

A former conjecture of Burr and Rosta [1], extending a conjecture of Erdős [2], asserted that in any two-colouring of the edges of a large complete graph, the proportion of subgraphs isomorphic to a fixed graphG which are monochromatic is at least the proportion found in a random colouring. It is now known that the conjecture fails for some graphsG, includingG=Kp forp≥4.We investigate for which graphsG the conjecture holds. Our main result is that the conjecture fails ifG containsK4 as a subgraph, and in particular it fails for almost all graphs.

ZERO MODES IN A SYSTEM OF AHARONOV–BOHM FLUXES

The quantum double is shown to imply the dressing transformation on quantum compact groups and the quantum Iwasawa decompositon in the general case. Quantum dressing orbits are described explicitly as *-algebras. The dual coalgebras consisting of differential operators are related to the quantum Weyl elements. Besides, the differential geometry on a quantum leaf allows a remarkably simple construction of irreducible *-representations of the algebras of quantum functions. Representation spaces then consist of analytic functions on classical phase spaces. These representations are also interpreted in the framework of quantization in the spirit of Berezin applied to symplectic leaves on classical compact groups. Convenient “coherent states” are introduced and a correspondence between classical and quantum observables is given.

Quantum line bundles on S2 and method of orbits for SUq(2)

We study the Floquet Hamiltonian -i∂t + H + V(ωt), acting in L2([0,T],ℋ, dt), as depending on the parameter ω = 2π/T. We assume that the spectrum of H in ℋ is discrete, , but possibly degenerate, and that t ↦ V(t) ∈ ℬ(ℋ) is a 2π-periodic function with values in the space of Hermitian operators on ℋ. Let J > 0 and set . Suppose that for some σ > 0 it holds true that ∑hm > hnMmMn (hm - hn)-σ < ∞ where Mm is the multiplicity of hm. We show that in that case there exist a suitable norm to measure the regularity of V, denoted ∊V, and positive constants, ∊⋆ and δ⋆, with the property: if ∊V < ∊⋆ then there exists a measurable subset Ω∞ ⊂ Ω0 such that its Lebesgue measure fulfills |Ω∞| ≥ |Ω0| - δ⋆ ∊V and the Floquet Hamiltonian has a pure point spectrum for all ω∈Ω∞.

Zero modes in a system of Aharonov–Bohm solenoids on the Lobachevsky plane

This review paper is devoted to topological global aspects of quantal description. The treatment concentrates on quantizations of kinematical observables — generalized positions and momenta. A broad class of quantum kinematics is rigorously constructed for systems, the configuration space of which is either a homogeneous space of a Lie group or a connected smooth finite-dimensional manifold without boundary. The class also includes systems in an external gauge field for an Abelian or a compact gauge group. Conditions for equivalence and irreducibility of generalized quantum kinematics are investigated with the aim of classification of possible quantizations. Complete classification theorems are given in two special cases. It is attempted to motivate the global approach based on a generalization of imprimitivity systems called quantum Borel kinematics. These are classified by means of global invariants — quantum numbers of topological origin. Selected examples are presented which demonstrate the richness of applications of Borel quantization. The review aims to provide an introductory survey of the subject and to be sufficiently selfcontained as well, so that it can serve as a standard reference concerning Borel quantization for systems admitting localization on differentiable manifolds.