Sufficient Conditions for an Attractive Potential to Possess Bound States. II

1965 ◽  
Vol 6 (7) ◽  
pp. 1105-1107 ◽  
Author(s):  
F. Calogero
Keyword(s):  
Bound States ◽  
Sufficient Conditions ◽  
Attractive Potential

PHASE SEPARATION AND FFLO PHASES IN ULTRACOLD GAS OF S = 5/2 ATOMS WITH ATTRACTIVE POTENTIAL IN A ONE-DIMENSIONAL TRAP

2012 ◽  
Vol 26 (16) ◽  
pp. 1230009 ◽  
Author(s):  
P. SCHLOTTMANN ◽  
A. A. ZVYAGIN
Keyword(s):  
Phase Separation ◽  
Bethe Ansatz ◽  
Bound States ◽  
Local Density ◽  
Attractive Potential ◽  
Fflo State ◽  
Effective Spin ◽  
One Dimensional ◽  
Bethe Ansatz Solution ◽  
The Rich

In the context of ultracold atoms with effective spin S = 5/2 confined to an elongated trap we study the one-dimensional Fermi gas interacting via an attractive δ-function potential using the Bethe ansatz solution. There are N = 2S + 1 = 6 fundamental states: The particles can either be unpaired or clustered in bound states of 2, 3, …, 2S and 2S + 1 fermions. The rich ground state phase diagram consists of these six states and various mixed phases in which combinations of the fundamental states coexist. Possible scenarios for phase separation due to the harmonic confinement along the tube are explored within the local density approximation. In an array of tubes with weak Josephson tunneling superfluid order may arise. The response functions determining the type of superfluid order are calculated using conformal field theory and the exact Bethe ansatz solution. They consist of a power law with distance times a sinusoidal term oscillating with distance. The wavelength of the oscillations is related to the periodicity of a generalized Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state.

scholarly journals COULOMB SCATTERING IN NON-COMMUTATIVE QUANTUM MECHANICS

2013 ◽  
Vol 53 (5) ◽  
pp. 427-432 ◽  
Author(s):  
Veronika Gáliková ◽  
Peter Prešnajder
Keyword(s):  
Bound States ◽  
Principal Quantum Number ◽  
Negative Energy ◽  
Attractive Potential ◽  
Coulomb Scattering ◽  
Coulomb Problem ◽  
High Energies ◽  
Full Solution ◽  
Energy Plane ◽  
Rotationally Invariant

Recently we formulated the Coulomb problem in a rotationally invariant NC configuration space specified by NC coordinates <em>x<sub>i</sub>, i</em> = 1, 2, 3, satisfying commutation relations<em> [x<sub>i</sub>, x<sub>j</sub> ] = 2iλε<sub>ijk</sub>x<sub>k</sub></em> (<em>λ</em> being our NC parameter). We found that the problem is exactly solvable: first we gave an exact simple formula for the energies of the negative bound states <em>E<sup>λ</sup><sub>n</sub></em> &lt; 0 (n being the principal quantum number), and later we found the full solution of the NC Coulomb problem. In this paper we present an exact calculation of the NC Coulomb scattering matrix <em>S<sup>λ</sup><sub>j</sub> (E)</em> in the <em>j</em>-th partial wave. As the calculations are exact, we can recognize remarkable non-perturbative aspects of the model: 1) energy cut-off — the scattering is restricted to the energy interval 0 &lt; <em>E</em> &lt; <em>E</em><sub>crit</sub> = 2/<em>λ</em><sup>2</sup>; 2) the presence of two sets of poles of the S-matrix in the complex energy plane — as expected, the poles at negative energy <em>E</em><sup>I</sup><sub><em>λ</em>n</sub> = <em>E</em><sup><em>λ</em></sup><sub>n</sub> for the Coulomb attractive potential, and the poles at ultra-high energies <em>E</em><sup>II</sup><sub><em>λ</em>n</sub> = <em>E</em><sub>crit</sub> − <em>E<sup>λ</sup></em><sub>n</sub> for the Coulomb <em>repulsive</em> potential. The poles at ultra-high energies disappear in the commutative limit <em>λ</em>→0.

Sufficient conditions for the existence of bound states of N particles with attractive potentials

1984 ◽  
Vol 100 (9) ◽  
pp. 460-462 ◽  
Author(s):  
F.A.B. Coutinho ◽  
C.P. Malta ◽  
J.Fernando Perez
Keyword(s):  
Bound States ◽  
Sufficient Conditions

Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with 𝒫𝒯-symmetry

2019 ◽  
Vol 150 (1) ◽  
pp. 171-204
Author(s):  
Tomáš Dohnal ◽  
Dmitry Pelinovsky
Keyword(s):  
Bound States ◽  
Periodic Potential ◽  
Sufficient Conditions ◽  
Amplitude Equation ◽  
One Dimension ◽  
Real Interval ◽  
Pitaevskii Equation ◽  
Spectral Bands ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Small Imaginary Part

AbstractThe stationary Gross–Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the 𝒫𝒯 (parity-time reversal) symmetry. Under rather general assumptions on the potentials, we prove bifurcations of 𝒫𝒯-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrödinger operator with a complex 𝒫𝒯-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrödinger equation. In addition, we provide sufficient conditions for the appearance of complex spectral bands when the complex 𝒫𝒯-symmetric potential has an asymptotically small imaginary part.

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