Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1203.1189

The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions

David Krejcirik·Helena Sedivakova
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Dirichlet Laplacian in a curved three-dimensional tube built along a spatial (bounded or unbounded) curve is investigated in the limit when the uniform cross-section of the tube diminishes. Both deformations due to bending and twisting of the tube are considered. We show that the Laplacian converges in a norm-resolvent sense to the well known one-dimensional Schroedinger operator whose potential is expressed in terms of the curvature of the reference curve, the twisting angle and a constant measuring the asymmetry of the cross-section. Contrary to previous results, we allow the reference curves to have non-continuous and possibly vanishing curvature. For such curves, the distinguished Frenet frame standardly used to define the tube need not exist and, moreover, the known approaches to prove the result for unbounded tubes do not work. Our main ideas how to establish the norm-resolvent convergence under the minimal regularity assumptions are to use an alternative frame defined by a parallel transport along the curve and a refined smoothing of the curvature via the Steklov approximation.

Explore related subjects

math.SPmath-phmath.APmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

David Krejcirik, Helena Sedivakova. 2012-03-06. The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions. https://doi.org/10.1142/s0129055x12500183

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Schrödinger operators with accretive potentials in weighted spaces

We analyse Schrödinger operators with accretive potentials in weighted spaces. We find conditions on potentials and weights for which the Dirichlet realisation, introduced by generalised form methods, has non-empty resolvent set. We establish a domain and graph norm separation property, as well as sufficient conditions for the compactness and Schatten class of the resolvent. Moreover, we investigate the relation between discrete spectra and eigenfunctions of operators in standard and weighted spaces. As applications we extend results on the completeness of eigensystems of operators with accretive potentials from standard to weighted spaces and analyse operator matrices exhibiting a Schur dominance property, in particular, related to a wave equation with strong accretive damping.

math.SP↗
arXiv2026

Eisenstein scattering and Plancherel decomposition on cuspidal Bruhat-Tits quotients

For arithmetic quotients of Bruhat--Tits trees with finitely many cusps, we establish an explicit unitary correspondence between the spherical Eisenstein transform, with the Eisenstein series normalized by their constant terms, and the scattering transform of an associated Jacobi operator with finite core. Tracking the Haar measure, stabilizer weights, height coordinates, and cusp widths yields the Plancherel measure and shows that the absolutely continuous spectrum has multiplicity equal to the number of cusps. From a discrete Green identity we derive a matrix-valued Maass--Selberg formula for the Hermitian matrix $iS(θ)^*\partial_θS(θ)$, where $S(θ)$ is the scattering matrix. Its trace is determined by $\det S(θ)$, while the full matrix retains additional cusp-to-cusp information. After the corresponding change of normalization, the finite Schur complement obtained by eliminating the cusp rays agrees with the resonance matrix of Arends-Peterson-Weich. Using their resonance computations as input, we distinguish eigenvalues supported entirely in the finite core from poles of the scattering matrix. The Nagao and $Γ_0(T)$ quotients, together with a four-cusp quotient arising from an elliptic curve over $\mathbb F_3$, make the normalizations and matrix-valued conclusions explicit.

math.SP↗
arXiv2026

Spectral projectors of bisectorial Clifford operators and applications to the generalized gradient

We consider right-linear operators $T$ on a right Hilbert module $V$ over the Clifford algebra R_n, whose S-spectrum lies in an acute double sector. For these bisectorial operators, we introduce spectral projectors P_\pm associated with the two cones of the double sector. They decompose the Hilbert module into two submodules V=V_++V_-, and the bisectorial operator T into two sectorial operators T|_\pm. A crucial but non-trivial, cornerstone in this theory is the boundedness of the projectors P_\pm, which is, in turn connected to a bounded H^\infty-functional calculus of the operator T. We provide two practical criteria: either the squared operator admits a bounded H^\infty-functional calculus, or the operator is m-accretive. Finally, we apply these results to the gradient operator \nabla_a with nonconstant coefficients. For the particular gradient with constant coefficients, we are even able to derive explicit representations of the submodules V_\pm and the projectors P_\pm in Fourier space. Moreover, we identify the sign of the gradient operator with the Clifford-Hilbert transform. This sign plays a central role in the fractional powers of vector operators, which are used, for instance, in the non-local Fourier law of heat propagation.

math.SP↗