Pierre Duclos

CURVATURE-INDUCED BOUND STATES IN QUANTUM WAVEGUIDES IN TWO AND THREE DIMENSIONS

Dirichlet Laplacian on curved tubes of a constant cross section in two and three dimensions is investigated. It is shown that if the tube is non-straight and its curvature vanishes asymptotically, there is always a bound state below the bottom of the essential spectrum. An upper bound to the number of these bound states in thin tubes is derived. Furthermore, if the tube is only slightly bent, there is just one bound state; we derive its behaviour with respect to the bending angle. Finally, perturbation theory of these eigenvalues in any thin tube with respect to the tube radius is constructed and some open questions are formulated.

Bound States in Curved Quantum Layers

Abstract: We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a non-compact surface embedded in ℝ3. We suppose that the latter is endowed with the geodesic polar coordinates and that the layer has the hard-wall boundary. Under the assumption that the surface curvatures vanish at infinity we find sufficient conditions which guarantee the existence of geometrically induced bound states.

Floquet Hamiltonians with pure point spectrum

AbstractWe consider Floquet Hamiltonians of the type

Spectral stability under tunneling

K_F : = - i\partial _t + H_0 + \beta V(\omega t)

On the location of the resonances for schrodinger

, whereH0, a selfadjoint operator acting in a Hilbert space ℋ, has simple discrete spectrumE10 for a given α>0,t↦V(t) is 2π-periodic andr times strongly continuously differentiable as a bounded operator on ℋ, ω and β are real parameters and the periodic boundary condition is imposed in time. We show, roughly, that providedr is large enough, β small enough and ω non-resonant, then the spectrum ofKf is pure point. The method we use relies on a successive application of the adiabatic treatment due to Howland and the KAM-type iteration settled by Bellissard and extended by Combescure. Both tools are revisited, adjusted and at some points slightly simplified.

Adiabatically switched-on electrical bias and the Landauer-Buttiker formula

Abstract We show that under suitable non-trapping conditions on the potential V at energy E the Schrodinger operator H = − h 2 Δ + V exhibits no resonance in a neighbourhood of E for small h . We briefly discuss some critical values of E like thresholds or barrier tops.

Absolute Continuity in Periodic Thin Tubes and Strongly Coupled Leaky Wires

We study the spectral properties of multiple well Schrödinger operators on ℝn. We give in particular upper bounds on energy shifts due to tunnel effect and localization properties of wave packets. Our methods are based on Agmon type estimates for resolvents in classically forbidden regions and geometric perturbation theory. Our results are valid also for an infinite number of wells, arbitrary spectral type and in non-semi-classical regimes.

WEAKLY REGULAR FLOQUET HAMILTONIANS WITH PURE POINT SPECTRUM

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 nonresonant with the frequency of the system and satisfies a Diophantine-type estimate, or if it is resonant. Furthermore, we show that the KAM method employed in the nonresonant case produces uniform bounds on the growth of energy for driven systems.