Searcharxiv⌕ Search

arXiv subjects

Pavel Exner

Publications and source records attributed to Pavel Exner.

At least 199 records · Page 11Linked to original sources
arXiv2000

Waveguides coupled through a semitransparent barrier: a Birman-Schwinger analysis

The paper is devoted to a model of a mesoscopic system consisting of a pair of parallel planar waveguides separated by an infinitely thin semitransparent boundary modeled by a transverse delta interaction. We develop the Birman-Schwinger theory for the corresponding generalized Schroedinger operator. The spectral properties become nontrivial if the barrier coupling is not invariant with respect to longitudinal translations, in particular, there are bound states if the barrier is locally more transparent in the mean and the coupling parameter reaches the same asymptotic value in both directions along the guide axis. We derive the weak-coupling expansion of the ground-state eigenvalue for the cases when the perturbation is small in the supremum and the L^1-norms. The last named result applies to the situation when the support of the leaky part shrinks: the obtained asymptotics differs from that of a double guide divided by a pierced Dirichlet barrier. We also derive an upper bound on the number of bound states.

math-ph↗
arXiv2000

Berry phase for a potential well transported in a homogeneous magnetic field

We consider a two-dimensional particle of charge $e$ interacting with a homogeneous magnetic field perpendicular to the plane and a potential well which is transported along a closed loop in the plane. We show that a bound state corresponding to a simple isolated eigenvalue acquires at that Berry's phase equal to $2π{\rm sgn} e$ times the number of flux quanta through the oriented area encircled by the loop. We also argue that this is a purely quantum effect since the corresponding Hannay angle is zero.

quant-ph↗
arXiv2000

Wave Function Shredding by Sparse Quantum Barriers

We discuss a model in which a quantum particle passes through $δ$ potentials arranged in an increasingly sparse way. For infinitely many barriers we derive conditions, expressed in terms ergodic properties of wave function phases, which ensure that the point and absolutely continuous parts are absent leaving a purely singularly continuous spectrum. For a finite number of barriers, the transmission coefficient shows extreme sensitivity to the particle momentum with fluctuation in many different scales. We discuss a potential application of this behavior for erasing the information carried by the wave function.

quant-ph↗
arXiv2000

Anomalous Pauli electron states for magnetic fields with tails

We consider a two-dimensional electron with an anomalous magnetic moment, g>2, interacting with a nonzero magnetic field B perpendicular to the plane which gives rise to a flux F. Recent results about the discrete spectrum of the Pauli operator are extended to fields with the O(r^{-2-δ}) decay at infinity: we show that if |F| exceeds an integer N, there is at least N+1 bound states. Furthermore, we prove that weakly coupled bound states exist under mild regularity assumptions also in the zero flux case.

math-ph↗
arXiv1999

Berry phase in magnetic systems with point perturbations

We study a two-dimensional charged particle interacting with a magnetic field, in general non-homogeneous, perpendicular to the plane, a confining potential, and a point interaction. If the latter moves adiabatically along a loop the state corresponding to an isolated eigenvalue acquires a Berry phase. We derive an expression for it and evaluate it in several examples such as a homogeneous field, a magnetic whisker, a particle confined at a ring or in quantum dots, a parabolic and a zero-range one. We also discuss the behavior of the lowest Landau level in this setting obtaining an explicit example of the Wilczek-Zee phase for an infinitely degenerated eigenvalue.

quant-ph↗
arXiv1999

Edge currents in the absence of edges

We investigate a charged two-dimensional particle in a homogeneous magnetic field interacting with a periodic array of point obstacles. We show that while Landau levels remain to be infinitely degenerate eigenvalues, between them the system has bands of absolutely continuous spectrum and exhibits thus a transport along the array. We also compute the band functions and the corresponding probability current.

cond-mat↗
arXiv1999

Magnetic strip waveguides

We analyze the spectrum of the "local" Iwatsuka model, i.e. a two-dimensional charged particle interacting with a magnetic field which is homogeneous outside a finite strip and translationally invariant along it. We derive two new sufficient conditions for absolute continuity of the spectrum. We also show that in most cases the number of open spectral gaps of the model is finite. To illustrate these results we investigate numerically the situation when the field is zero in the strip being screened, e.g. by a superconducting mask.

cond-mat↗
arXiv1999

Point interactions in a tube

We discuss discuss spectral and scattering properties of a particle confined to a straight Dirichlet tube in $\mathbb{R}^3$ with a family of point interactions.

math-ph↗
arXiv1999

Optimal eigenvalues for some Laplacians and Schrödinger operators depending on curvature

This article is an expanded version of the plenary talk given by Evans Harrell at QMath98, a meeting in Prague, June 1998. We consider Laplace operators and Schrödinger operators with potentials containing curvature on certain regions of nontrivial topology, especially closed curves, annular domains, and shells. Dirichlet boundary conditions are imposed on any boundaries. Under suitable assumptions we prove that the fundamental eigenvalue is maximized when the geometry is round. We also comment on the use of coordinate transformations for these operators and mention some open problems.

math-ph↗
arXiv1998

Stability of driven systems with growing gaps, Quantum rings and Wannier ladders

We consider a quantum particle in a periodic structure submitted to a constant external electromotive force. The periodic background is given by a smooth potential plus singular point interactions and has the property that the gaps between its bands are growing with the band index. We prove that the spectrum is pure point--i.e. trajectories of wave packets lie in compact sets in Hilbert space-- if the Bloch frequency is non-resonant with the frequency of the system and satisfies a Diophantine type estimate, or if it is resonant. Furthermore it is shown that the KAM method employed in the non-resonant case produces uniform bounds on the growth of energy for driven systems.

math-ph↗
arXiv1998

On the number of particles which a curved quantum waveguide can bind

We discuss the discrete spectrum of N particles in a curved planar waveguide. If they are neutral fermions, the maximum number of particles which the waveguide can bind is given by a one-particle Birman-Schwinger bound in combination with the Pauli principle. On the other hand, if they are charged, e.g., electrons in a bent quantum wire, the Coulomb repulsion plays a crucial role. We prove a sufficient condition under which the discrete spectrum of such a system is empty.

math.FA↗
arXiv1997

Probability current tornado loops in three-dimensional scattering

We consider scattering of a three-dimensional particle on a finite family of delta potentials. For some parameter values the scattering wavenctions exhibit nodal lines in the form of closed loops, which may touch but do not entangle. The corresponding probability current forms vortical singularities around these lines; if the scattered particle is charged, this gives rise to magnetic flux loops. The conclusions extend to scattering on hard obstacles or smooth potentials.

quant-ph↗
arXiv1997

Evanescent modes in a multiple scattering factorization

We discuss differences between the exact S-matrix for scattering on serial structures and a known factorized expression constructed of single-element S-matrices. As an illustration, we use an exactly solvable model of a quantum wire with two point impurities.

cond-mat.mes-hall↗
arXiv1997

Bound-state asymptotic estimates for window-coupled Dirichlet strips and layers

We consider the discrete spectrum of the Dirichlet Laplacian on a manifold consisting of two adjacent parallel strips or planar layers coupled by a finite number N of windows in the common boundary. If the windows are small enough, there is just one isolated eigenvalue. We find upper and lower asymptotic bounds on the gap between the eigenvalue and the essential spectrum in the planar case, as well as for N=1 in three dimensions. Based on these results, we formulate a conjecture on the weak-coupling asymptotic behaviour of such bound states.

funct-an↗
arXiv1997

Magnetoresonances on a lasso graph

We consider a charged spinless quantum particle confined to a graph consisting of a loop to which a halfline lead is attached; this system is placed into a homogeneous magnetic field perpendicular to the loop plane. We derive the reflection amplitude and show that there is an infinite ladder of resonances; analyzing the resonance pole trajectories we show that half of them turn into true embedded eigenvalues provided the flux through the loop is an integer or halfinteger multiple of the flux unit $hc/e$. We also describe a general method to solve the scattering problem on graphs of which the present model is a simple particular case. Finally, we discuss ways in which a state localized initially at the loop decays.

quant-ph↗
arXiv1995

Weakly coupled states on branching graphs

We consider a Schrödinger particle on a graph consisting of $\,N\,$ links joined at a single point. Each link supports a real locally integrable potential $\,V_j\,$; the self--adjointness is ensured by the $\,δ\,$ type boundary condition at the vertex. If all the links are semiinfinite and ideally coupled, the potential decays as $\,x^{-1-ε}$ along each of them, is non--repulsive in the mean and weak enough, the corresponding Schrödinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the $\,δ\,$ coupling constant may be interpreted in terms of a family of squeezed potentials.

funct-an↗
arXiv1995

Band spectra of rectangular graph superlattices

We consider rectangular graph superlattices of sides l1, l2 with the wavefunction coupling at the junctions either of the delta type, when they are continuous and the sum of their derivatives is proportional to the common value at the junction with a coupling constant alpha, or the "delta-prime-S" type with the roles of functions and derivatives reversed; the latter corresponds to the situations where the junctions are realized by complicated geometric scatterers. We show that the band spectra have a hidden fractal structure with respect to the ratio theta := l1/l2. If the latter is an irrational badly approximable by rationals, delta lattices have no gaps in the weak-coupling case. We show that there is a quantization for the asymptotic critical values of alpha at which new gap series open, and explain it in terms of number-theoretic properties of theta. We also show how the irregularity is manifested in terms of Fermi-surface dependence on energy, and possible localization properties under influence of an external electric field. KEYWORDS: Schroedinger operators, graphs, band spectra, fractals, quasiperiodic systems, number-theoretic properties, contact interactions, delta coupling, delta-prime coupling.

cond-mat↗