Lifshitz-Tails and Non-Lifshitz-Tails for One-Dimensional Random Point Interactions

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.

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Non-Hermitian Star-shaped Quantum Graphs

Acta Polytechnica ◽

10.14311/1819 ◽

2013 ◽

Vol 53 (3) ◽

Author(s):

Miloslav Znojil

Keyword(s):

Numerical Results ◽

Quantum Graphs ◽

Point Interactions ◽

One Dimensional

A compact review is given, and a few new numerical results are added to the recent studies of the q-pointed one-dimensional star-shaped quantum graphs. These graphs are assumed endowedwith certain specific, manifestly non-Hermitian point interactions, localized either in the inner vertex or in all of the outer vertices and carrying, in the latter case, an interesting zero-inflow interpretation.

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On the spectrum of the Schrödinger Hamiltonian with a particular configuration of three one-dimensional point interactions

Reports on Mathematical Physics ◽

10.1016/s0034-4877(10)00004-2 ◽

2009 ◽

Vol 64 (3) ◽

pp. 367-393 ◽

Cited By ~ 13

Author(s):

Silvestro Fassari ◽

Fabio Rinaldi

Keyword(s):

Point Interactions ◽

One Dimensional

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A REMARK ON ONE-DIMENSIONAL MANY-BODY PROBLEMS WITH POINT INTERACTIONS

International Journal of Modern Physics B ◽

10.1142/s0217979200000601 ◽

2000 ◽

Vol 14 (07) ◽

pp. 721-727 ◽

Cited By ~ 10

Author(s):

SERGIO ALBEVERIO ◽

LUDWIK DABROWSKI ◽

SHAO-MING FEI

Keyword(s):

Singular Point ◽

Bound States ◽

Quantum Mechanical ◽

Contact Interactions ◽

Many Body ◽

Point Interactions ◽

One Dimensional ◽

Integrable Quantum ◽

Scattering Matrices ◽

Many Body Systems

The integrability of one-dimensional quantum mechanical many-body problems with general contact interactions is extensively studied. It is shown that besides the pure (repulsive or attractive) δ-function interaction there is another singular point interactions which gives rise to a new one-parameter family of integrable quantum mechanical many-body systems. The bound states and scattering matrices are calculated for both bosonic and fermionic statistics.

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