• Corpus ID: 236493736

Quantum Systems at The Brink -- Existence and Decay Rates of Bound States at Thresholds; Critical Potentials and dimensionality

  title={Quantum Systems at The Brink -- Existence and Decay Rates of Bound States at Thresholds; Critical Potentials and dimensionality},
  author={Dirk Hundertmark and Michal Jex and Markus Lange},
One of the crucial properties of a quantum system is the existence of bound states. In this paper we present a necessary and sufficient condition for the Schrödinger operator to have a zero energy bound state. In particular we show that the asymptotic behaviour of the potential is the crucial ingredient. We derive the necessary and sufficient conditions for existence and absence with respect to the dimension. Our results are sharp and show high dependence on dimension. 
Quantum Systems at The Brink: Helium-type systems
In the present paper we study two challenging problems for helium–type systems. Existence of eigenvalues at thresholds and the asymptotic behavior of the corresponding eigenfunctions. Since the usual


Rigorous conditions for the existence of bound states at the threshold in the two-particle case*
In the framework of non-relativistic quantum mechanics and with the help of Green's functions formalism, we study the behaviour of weakly bound states in a non-central two-particle potential as they
Decay properties of zero-energy resonances of multi-particle Schr\"odinger operators and why the Efimov effect does not exist for systems of $N\geq 4$ particles.
We consider $N$-body Schr\"odinger operators with a virtual level at the threshold of the essential spectrum. We show that in the case of $N\geq 3$ particles in dimension $n\geq3$ virtual levels
On the virtual level of two-body interactions and applications to three-body systems in higher dimensions
We consider a system of three particles in dimension 4 and higher interacting via short-range potentials, where the two-body Hamiltonians have a virtual level at the bottom of the essential spectrum.
Quantum Systems at The Brink. Existence and Decay Rates of Bound States at Thresholds; Helium
It is well known that $N$-electron atoms undergoes unbinding for a critical charge of the nucleus $Z_c$, i.e. the atom has eigenstates for the case $Z> Z_c$ and it has no bound states for $Z<Z_c$. In
Quantum systems at the brink
  • existence and decay rates of bound states at thresholds; helium, arXiv:1908.04883 , 25
  • 2019
Zero-Energy Bound State Decay for Non-local Schrödinger Operators
We consider solutions of the eigenvalue equation at zero energy for a class of non-local Schrödinger operators with potentials decreasing to zero at infinity. Using a path integral approach, we
On Non-local Variational Problems with Lack of Compactness Related to Non-linear Optics
A simple proof of the existence of solutions of the dispersion management and diffraction management equations for a zero average dispersion, respectively, diffraction are given as maximizers of non-linear and non-local variational problems which are invariant under a large non-compact group.
Schrödinger Semigroups
Let H = \L + V be a general Schrödinger operator on R" (v~> 1), where A is the Laplace differential operator and V is a potential function on which we assume minimal hypotheses of growth and
Sharp counterexamples in unique continuation for second order elliptic equations
We construct nontrivial solutions with compact support for the elliptic equation ∆u = V u with V ∈ Lp, p < n/2 or V ∈ L w for n ≥ 3 and with V ∈ L1 for n = 2. The same method also yields nontrivial
Sharp condition on the decay of the potential for the absence of a zero-energy ground state of the Schrodinger equation
The authors prove a sharp criterion on the decay of the potential of a Schrodinger operator on R3 that ensures the absence of a zero-energy ground state. This condition complements results due to