Finite Number Of Gaps Research Articles

Absolute Continuity of the Spectrum for Periodically Modulated Leaky Wires in $${\mathbb{R}^{3}}$$

We consider a model of leaky quantum wire in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that such a system has a purely absolutely continuous spectrum and its negative part has band structure with an at most finite number of gaps. This result is extended also to the situation when there is an infinite number of the lines supporting the perturbations arranged periodically in one direction.

On spectral properties of periodic polyharmonic matrix operators

We consider a matrix operatorH = (-Δ)l +V inRn, wheren ≥ 2,l ≥ 1, 4l > n + 1, andV is the operator of multiplication by a periodic inx matrixV(x). We study spectral properties ofH in the high energy region. Asymptotic formulae for Bloch eigenvalues and the corresponding spectral projections are constructed. The Bethe-Sommerfeld conjecture, stating that the spectrum ofH can have only a finite number of gaps, is proved.

The Band Structure of the General Periodic¶ Schrödinger Operator with Point Interactions

The spectrum of Schrodinger operators H with periodic point potentials in dimensions d= 2, 3 is studied. In the general case of N points in the Wigner–Seitz cell it is proven that H has a band structure with at most a finite number of gaps (Bethe–Sommerfeld conjecture). It is also proven that in the case of a generic local point perturbation no singular continuous components are present; in the non-local case a fractal component like the Cantor set is exhibited, this component can either consist of a singular continuous or a dense point spectrum.

QUANTUM HALL EFFECT ON THE HYPERBOLIC PLANE IN THE PRESENCE OF DISORDER

We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in Comm. Math. Phys. 190 (1998), 629–673, to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian.

Band-gap structure of the spectrum of periodic Maxwell operators

We investigate the band-gap structure of some second-order differential operators associated with the propagation of waves in periodic two-component media. Particularly, the operator associated with the Maxwell equations with position-dependent dielectric constant e(x), x e R 3, is considered. The medium is assumed to consist of two components: the background, where e(x)= eh, and the embedded component composed of periodically positioned disjoint cubes, where ~(x) = r.,,. We show that the spectrum of the relevant operator has gaps provided some reasonable conditions are imposed on the parameters of the medium. Particularly, we show that one can open up at least one gap in the spectrum at any preassigned point 2 provided that the size of cubes L, the distance /= 6L between them, and the contrast e = eb/~,, are chosen in such a way that L--'~ 2, and quantities e- t6-3/2 and e62 are small enough. If these conditions are satisfied, the spectrum is located in a vicinity of width w~ (e.tS~/2) - ~ of the set {Tz2L-2k 2 : keZ3}. This means, in particular, that any finite number of gaps between the elements of this discrete set can be opened simultaneously, and the corresponding bands of the spectrum can be made arbitrarily narrow. The method developed shows that if the embedded component consists of periodically positioned balls or other domains which cannot pack the space without overlapping, one should expect pseudogaps rather than real gaps in the spectrum.

Wannier ladders and perturbation theory

Following Avron (1982) the authors consider the Stark effect for Bloch electrons in the case of a finite number of gaps. They prove that the ladders of resonances are given by the Wannier decoupled-band approximation and the perturbation theory. The Fermi golden rule yields the width behaviour of Buslaev and Dmitirieva (1990).

Modified Riccati approach to partially solvable quantum Hamiltonians. III. Related families of Pöschl-Teller potentials.

Within the modified Riccati approach developed elsewhere, we construct quasiexactly solvable potentials related to the P\"oschl-Teller and modified P\"oschl-Teller potentials. This is achieved by considering closed-form wave functions whose logarithmic derivatives are rational in the mapping u(x)=cosh(x) or u(x)=cos(x). With the hyperbolic mapping we construct an interesting four-parameter quasiexactly solvable confining potential with one, two, or three minima upon the value of its coupling constants. We give explicit expressions for the bound-state eigenfunctions and corresponding energy levels for some particular cases. The analytic continuation x\ensuremath{\rightarrow}ix transforms the previous potential into a periodic confining well, or a Kronig-Penney-like potential with an interesting structure of minima in each cell and a finite number of gaps. The energy levels in the last case are some of the band edges of the spectrum. In any case a finite but arbitrarily large number of energy levels and corresponding eigenfunctions are determined from the diagonalization of a finite matrix.

The lifetime of Wannier ladder states

I study the width of the Wannier ladder states, i.e, Bloch electrons in external homogeneous field. For periodic potentials with a finite number of gaps, a formula for the width is obtained showing that the width vanishes exponentially fast with the field in accordance with Zener tunneling. The case of infinitely many gaps is studied qualitatively, and it is argued that although the width decreases exponentially “on the average”, the detailed bahavior is very complicated. In particular the width oscillates over different orders of magnitude as the field changes slightly. The oscillations are a consequence of a resonance phenomenon.

Peierls-Fr�hlich problem and potentials with finite number of gaps. II

Peierls-Fr�hlich problem and potentials with finite number of gaps. II

On a periodic boundary value problem with a finite number of gaps in its spectrum

(--co, &>, (4 9 n;>, (A, 3 A>, t&T WY (3) are called instability intervals. All but the first interval in (3) are finite and may shrink to a point under special conditions. Erdelyi [3] discovered situations where all but a finite number of finite instability intervals vanish when q(t) is a suitable elliptic function. Lax [9] showed that a function q(t) which satisfies the nth order Korteweg-de Vries equation requires (1) to have no more than n non-vanishing finite instability intervals. Conversely, if all finite instability intervals vanish, then q(t) in (1) is necessarily a constant. This fact was proved by Borg [ 11, Hochstadt [ 71 and Ungar [ 121. When all but n finite instability intervals vanish q(t) satisfies a differential equation of the form

Peierls-Fr�hlich problem and potentials with finite number of gaps. I

Peierls-Fr�hlich problem and potentials with finite number of gaps. I

Periodic Multi-Soliton Solutions of Korteweg-de Vries Equation and Toda Lattice

A review of the recent works on the periodic multi-soliton solutions of the KdV equation is given. Connection with the theory of the abelian integrals is emphasized. Discrete analogue is also discussed leading to the exact solutions of the periodic Toda lattice. the works of Gardner, Greene, Kruskal and Miura (GG KM)ll and Lax. 2> On employing the inverse scattering theory, GG KM have found that the multi-soliton solutions, the proper generalization of the classical solitary wave solutions, can be constructed from the reflectionless potentials. After the work of Novikov3> it is established that the periodic analogue of the multi-soliton solutions is connected with the potentials which have finite number of gaps in the spectrum. An explicit realization of these potentials and associated solutions of the KdV equation is given by Dubrovin4l and Its-Matveev.5l Their method is based on the theory ·of the abelian integrals and goes back to Akhiezer.6l The present authors7> develop a discrete analogue of the construction of Dubrovin and Its-Matveev. To give a review of these works is the purpose of the present paper. As an independent line of development, Lax8> and then McKean­ Moerbeke9l have studied the periodic problem for the Sturm-Liouville equation and the KdV equation starting from the earlier work of Hochstadt10l and employing the Hamiltonian formalism for the higher order KdV equations. The relation with the theory of the abelian integrals is also indicated in Ref. 11). In §2 we describe generalities on the spectral properties of the Sturm­ Liouville equation with periodic potential. The potentials with finite number (say g) of gaps in the spectrum are studied. In §3 a hyperelliptic Riemann surface of genus g is introduced so that the Bloch eigenfunction is single-valued function on the surface. Then