On bound states for systems of weakly coupled Schrödinger equations in one space dimension

  title={On bound states for systems of weakly coupled Schr{\"o}dinger equations in one space dimension},
  author={Michael Melgaard},
  journal={Journal of Mathematical Physics},
  • M. Melgaard
  • Published 21 October 2002
  • Mathematics
  • Journal of Mathematical Physics
We establish the Birman–Schwinger relation for a class of Schrodinger operators −d2/dx2⊗1H+V on L2(R,H), where H is an auxiliary Hilbert space and V is an operator-valued potential. As an application we give an asymptotic formula for the bound states which may arise for a weakly coupled Schrodinger operator with a matrix potential (having one or more thresholds). In addition, for a two-channel system with eigenvalues embedded in the continuous spectrum we show that, under a small perturbation… 
2 Citations
Threshold properties of matrix-valued Schrödinger operators
We present some results on the perturbation of eigenvalues embedded at a threshold for a two-channel Hamiltonian with three-dimensional Schrodinger operators as entries and with a small off-diagonal


A class of analytic perturbations for one-body Schrödinger Hamiltonians
We study a class of symmetric relatively compact perturbations satisfying analyticity conditions with respect to the dilatation group inRn. Absence of continuous singular part for the Hamiltonians is
Scattering in one dimension: The coupled Schrödinger equation, threshold behaviour and Levinson’s theorem
We formulate scattering in one dimension due to the coupled Schrodinger equation in terms of the S matrix, the unitarity of which leads to constraints on the scattering amplitudes. Levinson’s theorem
Spectral Properties at a Threshold for Two-Channel Hamiltonians: II. Applications to Scattering Theory
Spectral properties and scattering theory in the low-energy limit are investigated for two-channel Hamiltonians with Schrodinger operators as component Hamiltonians. In various, mostly fairly
Spectral properties at a threshold for two-channel Hamiltonians I. Abstract theory
Spectral properties at thresholds are investigated for two-channel Hamiltonians in various, mostly fairly “singular” settings. In an abstract framework we deduce asymptotic expansions of the
v ANNOUNCEMENT: Special Issue on Quantum Information Theory QUANTUM PHYSICS; PARTICLES AND FIELDS 4627 Small-energy asymptotics of the scattering matrix for the matrix Schrodinger equation on the line
The one-dimensional matrix Schrodinger equation is considered when the matrix potential is self-adjoint with entries that are integrable and have finite first moments. The small-energy asymptotics of
Sharp Lieb-Thirring inequalities in high dimensions
We show how a matrix version of the Buslaev-Faddeev-Zakharov trace formulae for a one-dimensional Schr\"odinger operator leads to Lieb-Thirring inequalities with sharp constants $L^{cl}_{\gamma,d}$
Quantum Mechanics for Hamiltonians Defined as Quadratic Forms
This monograph combines a thorough introduction to the mathematical foundations of n-body Schrodinger mechanics with numerous new results to greatly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press.
Perturbation of eigenvalues embedded at a threshold
  • A. JensenM. Melgaard
  • Physics, Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2002
Results are obtained on perturbation of eigenvalues and half-bound states (zero-resonances) embedded at a threshold. The results are obtained in a two-channel framework for small off-diagonal
A simple proof of a theorem of Laptev and Weidl
A new and elementary proof of a recent result of Laptev and Weidl is given. It is a sharp Lieb-Thirring inequality for one dimensional Schroedinger operators with matrix valued potentials.