arXiv subjects

Matej Tusek

Publications and source records attributed to Matej Tusek.

6 recordsLinked to original sources

arXiv2017

Dispersionless wave packets in Dirac materials

We show that a wide class of quantum systems with translational invariance can host dispersionless, soliton-like, wave packets. We focus on the setting where the effective, two-dimensional Hamiltonian acquires the form of the Dirac operator. The proposed framework for construction of the dispersionless wave packets is illustrated on silicene-like systems with topologically nontrivial effective mass. Our analytical predictions are accompanied by a numerical analysis and possible experimental realizations are discussed.

cond-mat.mes-hall↗

arXiv2016

On an extension of the Iwatsuka model

We prove absolute continuity for an extended class of two-dimensional magnetic Hamiltonians that were initially studied by A. Iwatsuka. In particular, we add an electric field that is translation invariant in the same direction as the magnetic field is. As an example, we study the effective Hamiltonian for a thin quantum layer in a homogeneous magnetic field.

math-ph↗

arXiv2014

Atoms confined by very thin layers

The Hamiltonian of an atom with $N$ electrons and a fixed nucleus of infinite mass between two parallel planes is considered in the limit when the distance $a$ between the planes tends to zero. We show that this Hamiltonian converges in the norm resolvent sense to a Schrödinger operator acting effectively in $L^{2}(\mathbb{R}^{2N})$ whose potential part depends on $a$. Moreover, we prove that after an appropriate regularization this Schrödinger operator tends, again in the norm resolvent sense, to the Hamiltonian of a two-dimensional atom (with the three-dimensional Coulomb potential-one over distance), as $a\to 0$. This makes possible to locate the discrete spectrum of the full Hamiltonian once we know the spectrum of the latter one. Our results also provide a mathematical justification for the interest in the two-dimensional atoms with the three-dimensional Coulomb potential.

math-ph↗

arXiv2014

Nodal sets of thin curved layers

This paper is concerned with the location of nodal sets of eigenfunctions of the Dirichlet Laplacian in thin tubular neighbourhoods of hypersurfaces of the Euclidean space of arbitrary dimension. In the limit when the radius of the neighbourhood tends to zero, it is known that spectral properties of the Laplacian are approximated well by an effective Schrödinger operator on the hypersurface with a potential expressed solely in terms of principal curvatures. By applying techniques of elliptic partial differential equations, we strengthen the known perturbation results to get a convergence of eigenfunctions in Hölder spaces. This enables us in particular to conclude that every nodal set has a non-empty intersection with the boundary of the tubular neighbourhood.

math.AP↗

arXiv2013

The magnetic Laplacian in shrinking tubular neighbourhoods of hypersurfaces

The Dirichlet Laplacian between two parallel hypersurfaces in Euclidean spaces of any dimension in the presence of a magnetic field is considered in the limit when the distance between the hypersurfaces tends to zero. We show that the Laplacian converges in a norm-resolvent sense to a Schroedinger operator on the limiting hypersurface whose electromagnetic potential is expressed in terms of principal curvatures and the projection of the ambient vector potential to the hypersurface. As an application, we obtain an effective approximation of bound-state energies and eigenfunctions in thin quantum layers.

math-ph↗

arXiv2012

A New Estimate on the Two-Dimensional Indirect Coulomb Energy

We prove a new lower bound on the indirect Coulomb energy in two dimensional quantum mechanics in terms of the single particle density of the system. The new universal lower bound is an alternative to the Lieb--Solovej--Yngvason bound with a smaller constant, C=(4/3)^{3/2}\sqrt{5 π-1} \approx 5.90 < C_{LSY}= 192\sqrt{2 π} \approx 481.27, which also involves an additive gradient energy term of the single particle density.

math-ph↗