arXiv subjects

D. M. Gitman

Publications and source records attributed to D. M. Gitman.

At least 91 records · Page 5Linked to original sources

arXiv2011

Quantum motion in superposition of Aharonov-Bohm with some additional electromagnetic fields

The structure of additional electromagnetic fields to the Aharonov-Bohm field, for which the Schrödinger, Klein-Gordon, and Dirac equations can be solved exactly are described and the corresponding exact solutions are found. It is demonstrated that aside from the known cases (a constant and uniform magnetic field that is parallel to the Aharonov-Bohm solenoid, a static spherically symmetrical electric field, and the field of a magnetic monopole), there are broad classes of additional fields. Among these new additional fields we have physically interesting electric fields acting during a finite time, or localized in a restricted region of space. There are additional time-dependent uniform and isotropic electric fields that allow exact solutions of the Schrodinger equation. In the relativistic case there are additional electric fields propagating along the Aharonov-Bohm solenoid with arbitrary electric pulse shape.

quant-ph↗

arXiv2010

Symmetry preserving self-adjoint extensions of Schrödinger operators with singular potentials

We develop a general technique for finding self-adjoint extensions of a symmetric operator that respect a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general construction is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid.

math-ph↗

arXiv2010

Two and four-level systems in magnetic fields restricted in time

We describe some new exact solutions for two- and four-level systems. In all the cases, external fields have a restricted behavior in time. First, we consider two types of new solutions for one-spin equation, one of them is in a external magnetic field that acts during a finite time interval. A new solution for two interacting spins is found in the case when the field difference between the external fields in each spin vary adiabatically, vanishing on the time infinity. The latter system can be identified with a quantum gate realized by two coupled quantum dots. The probability of the Swap operation for such a gate can be explicitly expressed in terms of special functions. Using the obtained expressions, we construct plots for the Swap operation for some parameters of the external magnetic field and interaction function.

quant-ph↗

arXiv2010

Self-adjoint extensions and spectral analysis in the generalized Kratzer problem

We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional nonrelativistic motion of a particle in the potential field $V(x)=g_{1}x^{-1}+g_{2}x^{-2}$. For $g_{2}>0$ and $g_{1}<0$, the potential is known as the Kratzer potential and is usually used to describe molecular energy and structure, interactions between different molecules, and interactions between non-bonded atoms. We construct all self-adjoint Schrodinger operators with the potential $V(x)$ and represent rigorous solutions of the corresponding spectral problems. Solving the first part of the problem, we use a method of specifying s.a. extensions by (asymptotic) s.a. boundary conditions. Solving spectral problems, we follow the Krein's method of guiding functionals. This work is a continuation of our previous works devoted to Coulomb, Calogero, and Aharonov-Bohm potentials.

math-ph↗

arXiv2010

Quantum and pseudoclassical descriptions of nonrelativistic spinning particles in noncommutative space

We construct a nonrelativistic wave equation for spinning particles in the noncommutative space (in a sense, a $θ$-modification of the Pauli equation). To this end, we consider the nonrelativistic limit of the $θ$-modified Dirac equation. To complete the consideration, we present a pseudoclassical model (à la Berezin-Marinov) for the corresponding nonrelativistic particle in the noncommutative space. To justify the latter model, we demonstrate that its quantization leads to the $θ$-modified Pauli equation. Then, we extract $θ$-modified interaction between a nonrelativistic spin and a magnetic field from the $θ$-modified Pauli equation and construct a $θ$-modification of the Heisenberg model for two coupled spins placed in an external magnetic field. In the framework of such a model, we calculate the probability transition between two orthogonal EPR (Einstein-Podolsky-Rosen) states for a pair of spins in an oscillatory magnetic field and show that some of such transitions, which are forbidden in the commutative space, are possible due to the space noncommutativity.

hep-th↗

arXiv2010

Coherent states of non-relativistic electron in magnetic-solenoid field

We construct coherent states of a nonrelativistic electron in the magnetic-solenoid field, which is a superposition of the Aharonov-Bohm field and a collinear uniform magnetic field. In the problem under consideration there are two kind of coherent states, the first kind corresponds to classical trajectories which embrace the solenoid and the second one to trajectories which do not. Mean coordinates in the constructed coherent states are moving along classical trajectories, the coherent states maintain their form under the time evolution, and represent a complete set of functions, which can be useful in semi classical calculations. In the absence of the Aharonov-Bohm filed these states are reduced to the well-known in the case of uniform magnetic field Malkin-Man'ko coherent states.

quant-ph↗

arXiv2010

Quantization of the Damped Harmonic Oscillator Revisited

We return to the description of the damped harmonic oscillator by means of a closed quantum theory with a general assessment of previous works, in particular the Bateman-Caldirola-Kanai model and a new model recently proposed by one of the authors. We show the local equivalence between the two models and argue that latter has better high energy behavior and is naturally connected to existing open-quantum-systems approaches.

quant-ph↗

arXiv2010

Polarization rotation and Casimir effect in suspended graphene films

The low-energy quasi-excitations in graphene are known to be described as Dirac fermions in 2+1 dimensions. Adopting field-theoretical approach we investigate the interaction of these quasi-particles with 3+1 dimensional electromagnetic field focusing on the optical properties of suspended graphene layers and their Casimir interaction with ideal conductor. The magnitude of predicted effects (the rotation of polarization of light and the Casimir force) appears to be well within modern experimental capabilities.

cond-mat.mes-hall↗

arXiv2010

Classification of quantum relativistic orientable objects

Started from our work "Fields on the Poincare Group and Quantum Description of Orientable Objects" (EPJC,2009), we consider here a classification of orientable relativistic quantum objects in 3+1 dimensions. In such a classification, one uses a maximal set of 10 commuting operators (generators of left and right transformations) in the space of functions on the Poincare group. In addition to usual 6 quantum numbers related to external symmetries (given by left generators), there appear additional quantum numbers related to internal symmetries (given by right generators). We believe that the proposed approach can be useful for description of elementary spinning particles considering as orientable objects. In particular, their classification in the framework of the approach under consideration reproduces the usual classification but is more comprehensive. This allows one to give a group-theoretical interpretation to some facts of the existing phenomenological classification of known spinning particles.

hep-th↗

arXiv2009

Casimir interaction between a perfect conductor and graphene described by the Dirac model

We adopt the Dirac model for graphene and calculate the Casimir interaction energy between a plane suspended graphene sample and a parallel plane perfect conductor. This is done in two ways. First, we use the Quantum Field Theory (QFT) approach and evaluate the leading order diagram in a theory with 2+1 dimensional fermions interacting with 3+1 dimensional photons. Next, we consider an effective theory for the electromagnetic field with matching conditions induced by quantum quasi-particles in graphene. The first approach turns out to be the leading order in the coupling constant of the second one. The Casimir interaction for this system appears to be rather weak. It exhibits a strong dependence on the mass of the quasi-particles in graphene.

hep-th↗

arXiv2009

Self-adjoint Schrodinger and Dirac operators with Aharonov-Bohm and magnetic-solenoid fields

We study all the s.a. Schrodinger and Dirac operators (Hamiltonians) both with pure AB field and with magnetic-solenoid field. Then, we perform a complete spectral analysis for these operators, which includes finding spectra and spectral decompositions, or inversion formulas. In constructing the Hamiltonians and performing their spectral analysis, we respectively follow the von Neumann theory of s.a. extensions of symmetric differential operators and the Krein method of guiding functionals. The examples of similar consideration are given by us in arXiv:0903.5277, where a nonrelativistic particle in the Calogero potential field is considered and in Theor. Math. Phys. 150 (1) (2007) 34, where a Dirac particle in the Coulomb field of arbitrary charge is considered. However, due to peculiarities of the three-dimensional problems under consideration, we elaborated a generalization of the approach used in the study of the Dirac particle.

quant-ph↗

arXiv2009

On "Schwinger Mechanism for Gluon Pair Production in the Presence of Arbitrary Time Dependent Chromo-Electric Field"

Recently the paper "Schwinger Mechanism for Gluon Pair Production in the Presence of Arbitrary Time Dependent Chromo-Electric Field" by G. C. Nayak was published [Eur. Phys. J. C 59, 715 (2009); arXiv:0708.2430]. Its aim is to obtain an exact expression for the probability of non-perturbative gluon pair production per unit time per unit volume and per unit transverse momentum in an arbitrary time-dependent chromo-electric background field. We believe that the obtained expression is open to question. We demonstrate its inconsistency on some well-known examples. We think that this is a consequence of using the so-called "shift theorem" [arXiv:hep-th/0609192] in deriving the expression for the probability. We make some critical comments on the theorem and its applicability to the problem in question.

hep-th↗

arXiv2009

Gauge invariance and classical dynamics of noncommutative particle theory

We consider a model of classical noncommutative particle in an external electromagnetic field. For this model, we prove the existence of generalized gauge transformations. Classical dynamics in Hamiltonian and Lagrangian form is discussed, in particular, the motion in the constant magnetic field is studied in detail.

math-ph↗

arXiv2009

Large Oscillator representations for self-adjoint Calogero Hamiltonians

In the article arXiv:0903.5277 [quant-ph], we have presented a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential $V(x)=αx^{-2}$. In such a way, we have described all possible s.a. operators (s.a. Hamiltonians) associated with the formal differential expression $\check{H}=-d_{x}^{2}+αx^{-2}$ for the Calogero Hamiltonian. Here, we discuss a new aspect of the problem, the so-called oscillator representation for the Calogero Hamiltonians. As it is know, operators of the form $\hat{N}=\hat{a}^{+}\hat{a}$ and $\hat{A}=\hat{a}\hat{a}^{+}$ are called operators of oscillator type. Oscillator type operators obey several useful properties in case if the elementary operator $\hat{a}$ and $\hat{a}^{+}$ are densely defined. It turns out that some s.a. Calogero Hamiltonians are oscillator type operators. We describe such Hamiltonians and find the corresponding mutually adjoint elementary operators.

quant-ph↗

arXiv2009

Dirac Equation in Noncommutative Space for Hydrogen Atom

We consider the energy levels of a hydrogen-like atom in the framework of $θ$-modified, due to space noncommutativity, Dirac equation with Coulomb field. It is shown that on the noncommutative (NC) space the degeneracy of the levels $2S_{1/2}, 2P_{1/2}$ and $ 2P_{3/2}$ is lifted completely, such that new transition channels are allowed.

hep-th↗

arXiv2009

Semiclassical and quantum description of motion on noncommutative plane

We study the canonical and the coherent state quantization of a particle moving in a magnetic field on a non-commutative plane. Starting from the so called θ-modified action, we perform the canonical quantization and analyze the gauge dependence of the obtained quantum theory. We construct the Malkin-Man'ko coherent states of the system in question, and the corresponding quantization. On this base, we study the relation between the coherent states and the "classical" trajectories predicted by the θ-modified action. In addition, we construct different semiclassical states, making use of special properties of circular squeezed states. With the help of these states, we perform the Berezin-Klauder-Toeplitz quantization and present a numerical exploration of the semiclassical behavior of physical quantities in these states.

quant-ph↗

arXiv2009

Coherent states of a particle in magnetic field and Stieltjes moment problem

A solution to a version of the Stieltjes moment problem is presented. Using this solution, we construct a family of coherent states of a charged particle in a uniform magnetic field. We prove that these states form an overcomplete set that is normalized and solves the identity. By the help of the coherent states we construct the Fock-Bergmann representation related to the particle quantization. This quantization procedure takes into account a circle topology of the classical motion.

quant-ph↗

arXiv2009

Self-adjoint extensions and spectral analysis in Calogero problem

In this paper, we present a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential $αx^{-2}$. Although the problem is quite old and well-studied, we believe that our consideration, based on a uniform approach to constructing a correct quantum-mechanical description for systems with singular potentials and/or boundaries, proposed in our previous works, adds some new points to its solution. To demonstrate that a consideration of the Calogero problem requires mathematical accuracy, we discuss some "paradoxes" inherent in the "naive" quantum-mechanical treatment. We study all possible self-adjoint operators (self-adjoint Hamiltonians) associated with a formal differential expression for the Calogero Hamiltonian. In addition, we discuss a spontaneous scale-symmetry breaking associated with self-adjoint extensions. A complete spectral analysis of all self-adjoint Hamiltonians is presented.

quant-ph↗