Searcharxiv⌕ Search

arXiv subjects

J. Dittrich

Publications and source records attributed to J. Dittrich.

13 recordsLinked to original sources
arXiv2024

Dirac particle under dynamical confinement: Fermi acceleration, trembling motion and quantum force

Quantum dynamics of a Dirac particle in a 1D box with moving wall is studied. Dirac equation with time-dependent boundary condition is mapped onto that with static one, but with time-dependent mass. Exact analytical solution of such modified Dirac equation is obtained for massless particle. For massive particle the problem is solved numerically. Time-dependences of the main characteristics of the dynamical confinement, such as average kinetic energy and quantum force are analyzed. It is found that the average kinetic energy remains bounded for the interval length bounded from below, in particular for the periodically oscillating wall.

quant-ph↗
arXiv2020

Scattering of particles bounded to an infinite planar curve

Non-relativistic quantum particles bounded to a curve in R^2 by attractive contact $δ$-interaction are considered. The interval between the energy of the transversal bound state and zero is shown to belong to the absolutely continuous spectrum, with possible embedded eigenvalues. The existence of the wave operators is proved for the mentioned energy interval using the Hamiltonians with the interaction supported by the straight lines as the free ones. Their completeness is not proved. The curve is assumed C^3-smooth, non-intersecting, unbounded, asymptotically approaching two different half-lines (non-parallel or parallel but excluding the "U-case"). Physically, the system can be considered as a model of long nanostructural channel.

math-ph↗
arXiv2017

On the solution to the separated equation in the 3-particle Calogero-Moser problem

We propose the exact solution of the equation in separated variable which appears in the process of constructing solutions to the quantum Calogero-Moser three-particle problem with elliptic two-particle potential $g(g-1)\wp(q)$. This solution is found for special values of coupling constants $g\in {\mathbb Z}, \, g>1$. It can be used for solving three-paricle CM problem under appropriate boundary conditions.

math-ph↗
arXiv2015

On eigenvalue asymptotics for strong delta-interactions supported by surfaces with boundaries

Let $S\subset\mathbb{R}^3$ be a $C^4$-smooth relatively compact orientable surface with a sufficiently regular boundary. For $β\in\mathbb{R}_+$, let $E_j(β)$ denote the $j$th negative eigenvalue of the operator associated with the quadratic form \[ H^1(\mathbb{R}^3)\ni u\mapsto \iiint_{\mathbb{R}^3} |\nabla u|^2dx -β\iint_S |u|^2dσ, \] where $σ$ is the two-dimensional Hausdorff measure on $S$. We show that for each fixed $j$ one has the asymptotic expansion \[ E_j(β)=-\dfrac{β^2}{4}+μ^D_j+ o(1) \;\text{ as }\; β\to+\infty\,, \] where $μ_j^D$ is the $j$th eigenvalue of the operator $-Δ_S+K-M^2$ on $L^2(S)$, in which $K$ and $M$ are the Gauss and mean curvatures, respectively, and $-Δ_S$ is the Laplace-Beltrami operator with the Dirichlet condition at the boundary of $S$. If, in addition, the boundary of $S$ is $C^2$-smooth, then the remainder estimate can be improved to ${\mathcal O}(β^{-1}\logβ)$.

math-ph↗
arXiv2014

Scattering through a straight quantum waveguide with combined boundary conditions

Scattering through a straight two-dimensional quantum waveguide Rx(0,d) with Dirichlet boundary conditions on (-\infty,0)x{y=0} \cup (0,\infty)x{y=d} and Neumann boundary condition on (-infty,0)x{y=d} \cup (0,\infty)x{y=0} is considered using stationary scattering theory. The existence of a matching conditions solution at x=0 is proved. The use of stationary scattering theory is justified showing its relation to the wave packets motion. As an illustration, the matching conditions are also solved numerically and the transition probabilities are shown.

math-ph↗
arXiv2003

Analytic proof of the Sutherland conjecture

Using the integral representation of the inverse of the logarithmic derivative of the elliptic theta function, the spectrum of the Lax matrix for the 1D system of particles interacting via inverse sinh-squared potential is shown to be given by the asymptotic Bethe ansatz in the thermodynamic limit.

math-ph↗
arXiv2002

Nonsmoothness of the boundary and the relevant heat kernel coefficients

The contributions to the heat kernel coefficients generated by the corners of the boundary are studied. For this purpose the internal and external sectors of a wedge and a cone are considered. These sectors are obtained by introducing, inside the wedge, a cylindrical boundary. Transition to a cone is accomplished by identification of the wedge sides. The basic result of the paper is the calculation of the individual contributions to the heat kernel coefficients generated by the boundary singularities. In the course of this analysis certain patterns, that are followed by these contributions, are revealed. The implications of the obtained results in calculations of the vacuum energy for regions with nonsmooth boundary are discussed. The rules for obtaining all the heat kernel coefficients for the minus Laplace operator defined on a polygon or in its cylindrical generalization are formulated.

hep-th↗
arXiv2002

Massive Scalar Field in an One-Dimensional Oscillating Region

The classical scalar massive field satisfying the Klein-Gordon equation in a finite one-dimensional space interval of periodically varying length with Dirichlet boundary conditions is studied. For the sufficiently small mass, the energy can exponentially grow with time under the same conditions as for the massless case. The proofs are based on estimates of exactly given mass-induced corrections to the massless case.

math-ph↗
arXiv2000

On the Ground State of Ferromagnetic Hamiltonians

It is generally believed that the ground state of the ferromagnetic Heisenberg-Dirac-Van Vleck Hamiltonians acting on s=1/2 spins of a lattice with N sites has the maximal possible value of the total spin S=N/2 and is N+1 times degenerate. We present a rigorous proof of this statement, independent of the lattice dimension and topology.

math-ph↗
arXiv2000

A model of interband radiative transitions

We consider a simple model which is a caricature of a crystal interacting with a radiation field. The model has two bands of continuous spectrum and the particle can pass from the upper one to the lower by radiating a photon, the coupling between the excited and deexcited states being of a Friedrichs type. Under suitable regularity and analyticity assumptions we find the continued resolvent and show that for weak enough coupling it has a curve-type singularity in the lower halfplane which is a deformation of the upper-band spectral cut. We then find a formula for the decay amplitude and show that for a fixed energy it is approximately exponential at intermediate times, while the tail has a power-like behaviour.

math-ph↗
arXiv1997

On the second-neighbor correlator in 1D XXX quantum antiferromagnetic spin chain

We have calculated the energy per site for the ground state of antiferromagnetic quantum spin chain with variable range exchange $h(j-k)\propto \sinh^2 a \sinh^{-2}a(j-k)$ in the framework of the asymptotic Bethe ansatz. By expanding it in powers of $e^{-2a}$, we have confirmed the value of the second-neighbor correlator for the model with nearest-neighbor exchange obtained earlier in the atomic limit of the Hubbard chain.

cond-mat.stat-mech↗
arXiv1996

On the two-magnon bound states for the quantum Heisenberg chain with variable range exchange

The spectrum of finite-difference two-magnon operator is investigated for quantum S=1/2 chain with variable range exchange of the form $h(j-k)\propto \sinh^{-2}a(j-k)$. It is found that usual bound state appears for some values of the total pseudomomentum of two magnons as for the Heisenberg chain with nearest-neighbor spin interaction. Besides this state, a new type of bound state with oscillating wave function appears at larger values of the total pseudomomentum.

solv-int↗