Quadratic Commutation Relations Research Articles

Semiclassical Asymptotics of the Spectrum of the Hydrogen Atom in an Electromagnetic Field Near the Upper Boundaries of Spectral Clusters

We study the Zeeman–Stark effect in the hydrogen atom located in an electromagnetic field by using irreducible representations of an algebra with the Karasev–Novikova quadratic commutation relations. The representations are associated with resonance spectral clusters near the energy level of the unperturbed hydrogen atom. We find asymptotics for a series of eigenvalues and corresponding asymptotic eigenfunctions near the upper boundaries of spectral clusters in the case of positive intensities of the electric field.

Asymptotics of the Spectrum of the Hydrogen Atom in Orthogonal Electric and Magnetic Fields near the Lower Boundaries of Spectral Clusters

The Zeeman—Stark effect for the hydrogen atom in an electromagnetic field is considered by using irreducible representations of an algebra with Karasev—Novikova quadratic commutation relations. The asymptotics of the series of eigenvalues and asymptotic eigenfunctions are obtained near the lower boundaries of the resonance spectral clusters, which are formed near the energy levels of the unperturbed hydrogen atom.

Tensor Bogolyubov representations of the renormalized square of white noise (RSWN) algebra

We introduce the quadratic analog of the tensor Bogolyubov representation of the CCR. Our main result is the determination of the structure of these maps: each of them is uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text] into itself whose inverses induce transformations that map the Lebesgue measure [Formula: see text] into measures [Formula: see text] absolutely continuous with respect to it. Furthermore, the Radon–Nikodyn derivatives [Formula: see text], of these measures with respect to [Formula: see text], must satisfy the relation [Formula: see text] for [Formula: see text]-almost every [Formula: see text]. This makes a surprising bridge with the hyperbolic sine and cosine defining the structure of usual (i.e. first-order) Bogolyubov transformations. The reason of the surprise is that the linear and quadratic commutation relations are completely different.

On the Asymptotics of the Spectrum of the Hydrogen Atom in Orthogonal Electric and Magnetic Fields Near the Upper Boundaries of Spectral Clusters

The problem of the Zeemann-Stark effect for the hydrogen atom in electromagnetic fields is considered using the irreducible representations of the Karasev-Novikova algebra with quadratic commutation relations. An asymptotics of the series of eigenvalues and the asymptotic eigenfunctions are obtained near the upper boundaries of resonance spectral clusters which are formed near the energy levels of an unperturbed hydrogen atom.

Dolbeault–Dirac Operators, Quantum Clifford Algebras and the Parthasarathy Formula

We consider Dolbeault-Dirac operators on quantized irreducible flag manifolds as defined by Kr\"ahmer and Tucker-Simmons. We show that, in general, these operators do not satisfy a formula of Parthasarathy-type. This is a consequence of two results that we prove here: we always have quadratic commutation relations for the relevant quantum root vectors, up to terms in the quantized Levi factor; there are examples of quantum Clifford algebras where the commutation relations are not of quadratic-constant type.

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

We investigate the second-order Zeeman effect in a magnetic field using irreducible representations of an algebra with the Karasev - Novikova quadratic commutation relations. To each such representation there corresponds a spectral cluster near the energy level of the unperturbed hydrogen atom. Using this model as an example, we describe a general method for constructing asymptotic solutions near the boundaries of spectral clusters based on a new integral representation. We also study the problem of computing quantum averages near the lower boundaries of clusters.

Algebra with Quadratic Commutation Relations for an Axially Perturbed Coulomb–Dirac Field

The motion of a particle in the field of an electromagnetic monopole (in the Coulomb–Dirac field) perturbed by an axially symmetric potential after quantum averaging is described by an integrable system. Its Hamiltonian can be written in terms of the generators of an algebra with quadratic commutation relations. We construct the irreducible representations of this algebra in terms of second-order differential operators; we also construct its hypergeometric coherent states. We use these states in the first-order approximation with respect to the perturbing field to obtain the integral representation of the eigenfunctions of the original problem in terms of solutions of the model Heun-type second-order ordinary differential equation and present the asymptotic approximation of the corresponding eigenvalues.

Representation of exact and semiclassical eigenfunctions via coherent states. Hydrogen atom in a magnetic field

Global formulas for eigenfunctions and solutions to the Cauchy problem, including the path integral representation, are obtained using the coherent states technique. The reduction of coherent states via symmetry groups is studied for a transformation from “Bessel” to “hypergeometric” states. The eigenfunctions of the Hamiltonian for the hydrogen atom in a homogeneous magnetic field are expressed in terms of Bessel coherent states. For a small field, after quantum averaging, the Hamiltonian is represented in terms of generators with quadratic commutation relations. The irreducible representations of this quadratic algebra are realized on hypergeometric states. The notion of deformed hypergeometric states is also introduced for this quadratic algebra as an analog of squeezed Gaussian packets of the Heisenberg algebra. The asymptotic equations of eigenfunctions with respect to a small field and a large leading quantum number are derived using these states and their “deaveraging.” Some explicit formulas for the Zeeman splitting of the spectrum are obtained up to the fourth order with respect to the field, as well as for lower and upper levels in the cluster, including the case of “incidence on the center.”

Generic algebras with quadratic commutation relations

Generic algebras with m≥3 generators and quadratic relations are completely described. A conjecture of A. M. Vershik about their finite-dimensionality is proved.