1D Schrödinger operators with short range interactions: two-scale regularization of distributional potentials

Abstract

For real bounded functions Φand Ψof compact support, we prove the norm resolvent convergence, as εand νtend to 0, of a family of one-dimensional Schroedinger operators on the line of the form S_{ε, ν}= -D^2+αε^{-2}Φ(ε^{-1}x)+βν^{-1}Ψ(ν^{-1}x), provided the ratio ν/εhas a finite or infinity limit. The limit operator S_0 depends on the shape of Φand Ψas well as on the limit of ratio ν/ε. If the potential αΦpossesses a zero-energy resonance, then S_0 describes a non trivial point interaction at the origin. Otherwise S_0 is the direct sum of the Dirichlet half-line Schroedinger operators.

20 pages, minor corrections

References in corpus (1)

• Coupling in the singular limit of thin quantum waveguides

Cited by in corpus (18)

• Green's function approach for quantum graphs: an overview
• Distribution theory for Schrödinger's integral equation
• Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials
• Single point potentials with total resonant tunneling
• Point interactions with bias potentials
• Schrödinger operators with singular rank-two perturbations and point interactions
• A phenomenon of splitting resonant-tunneling one-point interactions
• Two-parametric ${\delta }^{\prime }$-interactions: approximation by Schrödinger operators with localized rank-two perturbations
• Approximations of quantum-graph vertex couplings by singularly scaled potentials
• Scattering data and bound states of a squeezed double-layer structure
• Schrödinger operators with Coulomb-like potentials
• Bound states of a one-dimensional Dirac equation with multiple delta-potentials
• Bound states and point interactions of the one-dimensional pseudospin-one Hamiltonian
• Origin of resonant tunneling through single-point barriers
• On negative eigenvalues of 1D Schrödinger operators with ${\delta }^{\prime }$-like potentials
• Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
• 2D Schrödinger operators with singular potentials concentrated near curves
• Conditions for realizing one-point interactions from a multi-layer structure model