Erratum: A sufficient condition for the existence of bound states in a potential [J. Math. Phys. 38, 4900 (1997)]

2004 ◽  
Vol 45 (7) ◽  
pp. 2987-2987
Author(s):  
K. Chadan ◽  
R. Kobayashi
Keyword(s):  
Bound States ◽  
Sufficient Condition ◽  
Math Phys

scholarly journals The spectrum of the Schrödinger–Hamiltonian for trapped particles in a cylinder with a topological defect perturbed by two attractive delta interactions

2018 ◽  
Vol 15 (08) ◽  
pp. 1850135 ◽  
Author(s):  
Fassari Silvestro ◽  
Rinaldi Fabio ◽  
Viaggiu Stefano
Keyword(s):  
Quantum Dot ◽  
Harmonic Oscillator ◽  
Total Energy ◽  
Spectral Properties ◽  
Bound States ◽  
Three Dimensional ◽  
Point Interactions ◽  
One Dimensional ◽  
The One ◽  
Math Phys

In this paper, we exploit the technique used in [Albeverio and Nizhnik, On the number of negative eigenvalues of one-dimensional Schrödinger operator with point interactions, Lett. Math. Phys. 65 (2003) 27; Albeverio, Gesztesy, Hoegh-Krohn and Holden, Solvable Models in Quantum Mechanics (second edition with an appendix by P. Exner, AMS Chelsea Series 2004); Albeverio and Kurasov, Singular Perturbations of Differential Operators: Solvable Type Operators (Cambridge University Press, 2000); Fassari and Rinaldi, On the spectrum of the Schrödinger–Hamiltonian with a particular configuration of three one-dimensional point interactions, Rep. Math. Phys. 3 (2009) 367; Fassari and Rinaldi, On the spectrum of the Schrödinger–Hamiltonian of the one-dimensional harmonic oscillator perturbed by two identical attractive point interactions, Rep. Math. Phys. 3 (2012) 353; Albeverio, Fassari and Rinaldi, The Hamiltonian of the harmonic oscillator with an attractive-interaction centered at the origin as approximated by the one with a triple of attractive-interactions, J. Phys. A: Math. Theor. 49 (2016) 025302; Albeverio, Fassari and Rinaldi, Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive [Formula: see text]-impurities symmetrically situated around the origin II, Nanosyst. Phys. Chem. Math. 7(5) (2016) 803; Albeverio, Fassari and Rinaldi, Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive [Formula: see text]-impurities symmetrically situated around the origin, Nanosyst. Phys. Chem. Math. 7(2) (2016) 268] to deal with delta interactions in a rigorous way in a curved spacetime represented by a cosmic string along the [Formula: see text] axis. This mathematical machinery is applied in order to study the discrete spectrum of a point-mass particle confined in an infinitely long cylinder with a conical defect on the [Formula: see text] axis and perturbed by two identical attractive delta interactions symmetrically situated around the origin. We derive a suitable approximate formula for the total energy. As a consequence, we found the existence of a mixing of states with positive or zero energy with the ones with negative energy (bound states). This mixture depends on the radius [Formula: see text] of the trapping cylinder. The number of quantum bound states is an increasing function of the radius [Formula: see text]. It is also interesting to note the presence of states with zero total energy (quasi free states). Apart from the gravitational background, the model presented in this paper is of interest in the context of nanophysics and graphene modeling. In particular, the graphene with double layer in this framework, with the double layer given by the aforementioned delta interactions and the string on the [Formula: see text]-axis modeling topological defects connecting the two layers. As a consequence of these setups, we obtain the usual mixture of positive and negative bound states present in the graphene literature.

On a sufficient condition of the existence of N-particle bound states

1986 ◽  
Vol 19 (12) ◽  
pp. 2337-2344 ◽  
Author(s):  
D Bolle ◽  
K Chadan ◽  
G Karner
Keyword(s):  
Bound States ◽  
Sufficient Condition

On the number of bound states of semirelativistic Hamiltonian with Dirac delta potentials in one dimension

2018 ◽  
Vol 96 (11) ◽  
pp. 1235-1241
Author(s):  
Fatih Erman
Keyword(s):  
Bound States ◽  
Bound State ◽  
One Dimension ◽  
Sufficient Condition ◽  
Explicit Criterion ◽  
State Problem ◽  
Dirac Delta ◽  
Bound State Problem

We study the bound state problem for semirelativistic N attractive Dirac δ-potentials in one dimension. We give a sufficient condition for the Hamiltonian to have N bound states and give an explicit criterion for it.

scholarly journals A complete classification of threshold properties for one-dimensional discrete Schrödinger operators

2015 ◽  
Vol 27 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Kenichi Ito ◽  
Arne Jensen
Keyword(s):  
Bound States ◽  
Complete Classification ◽  
Full Generality ◽  
One Dimensional ◽  
All Solutions ◽  
Growth Properties ◽  
Discrete Schrödinger Operators ◽  
The One ◽  
Math Phys ◽  
Generalized Zero

We consider the one-dimensional discrete Schrödinger operator on ℤ, and study the relation between the generalized eigenstates and the asymptotic expansion of the resolvent for the threshold 0. We decompose the generalized zero eigenspace into subspaces, some of which correspond to the bound states or the resonance states, only by their growth properties at infinity, and precisely describe the first few leading coefficients in the expansion using these subspaces. The generalized zero eigenspace we consider is the largest possible one, consisting of all solutions to the eigenequation. For the resolvent expansion, we implement the expansion scheme of Jensen–Nenciu [Rev. Math. Phys. 13 (2001) 717–754] and [Rev. Math. Phys. 16 (2004) 675–677] in its full generality.

A sufficient condition for the existence of bound states in a potential

1997 ◽  
Vol 38 (10) ◽  
pp. 4900-4908 ◽  
Author(s):  
K. Chadan ◽  
R. Kobayashi
Keyword(s):  
Bound States ◽  
Sufficient Condition
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