scholarly journals Kasteleyn Theorem, Geometric Signatures and KP-II Divisors on Planar Bipartite Networks in the Disk

2021 ◽  
Vol 24 (4) ◽  
Author(s):  
Simonetta Abenda
Keyword(s):  
Wave Function ◽  
Boundary Conditions ◽  
Spectral Curve ◽  
Dual Graph ◽  
Soliton Solutions ◽  
Geometric Interpretation ◽  
Bipartite Networks ◽  
Negative Part ◽  
Weighted Version ◽  
Natural Parametrization

AbstractMaximal minors of Kasteleyn sign matrices on planar bipartite graphs in the disk count dimer configurations with prescribed boundary conditions, and the weighted version of such matrices provides a natural parametrization of the totally non–negative part of real Grassmannians (Postnikov et al. J. Algebr. Combin. 30(2), 173–191, 2009; Lam J. Lond. Math. Soc. (2) 92(3), 633–656, 2015; Lam 2016; Speyer 2016; Affolter et al. 2019). In this paper we provide a geometric interpretation of such variant of Kasteleyn theorem: a signature is Kasteleyn if and only if it is geometric in the sense of Abenda and Grinevich (2019). We apply this geometric characterization to explicitly solve the associated system of relations and provide a new proof that the parametrization of positroid cells induced by Kasteleyn weighted matrices coincides with that of Postnikov boundary measurement map. Finally we use Kasteleyn system of relations to associate algebraic geometric data to KP multi-soliton solutions. Indeed the KP wave function solves such system of relations at the nodes of the spectral curve if the dual graph of the latter represents the soliton data. Therefore the construction of the divisor is automatically invariant, and finally it coincides with that in Abenda and Grinevich (Sel. Math. New Ser. 25(3), 43, 2019; Abenda and Grinevich 2020) for the present class of graphs.

scholarly journals Investigating Quantum Coherence by Negative Excursions of the Wigner Quasi-Distribution

2019 ◽  
Vol 9 (7) ◽  
pp. 1344 ◽  
Author(s):  
Mauro Ballicchia ◽  
David Ferry ◽  
Mihail Nedjalkov ◽  
Josef Weinbub
Keyword(s):  
Wave Function ◽  
Quantum Interference ◽  
Quantum Coherence ◽  
Quantum Communication ◽  
Electron System ◽  
Physical Problem ◽  
Quantum Superposition ◽  
Negative Part ◽  
Initial Momentum ◽  
The Past

Quantum information and quantum communication are both strongly based on concepts of quantum superposition and entanglement. Entanglement allows distinct bodies, that share a common origin or that have interacted in the past, to continue to be described by the same wave function until evolution is coherent. So, there is an equivalence between coherence and entanglement. In this paper, we show the relation between quantum coherence and quantum interference and the negative parts of the Wigner quasi-distribution, using the Wigner signed-particle formulation. A simple physical problem consisting of electrons in a nanowire interacting with the potential of a repulsive dopant placed in the center of it creates a quasi two-slit electron system that separates the wave function into two entangled branches. The analysis of the Wigner quasi-distribution of this problem establishes that its negative part is principally concentrated in the region after the dopant between the two entangled branches, maintaining the coherence between them. Moreover, quantum interference is shown in this region both in the positive and in the negative part of the Wigner function and is produced by the superposition of Wigner functions evaluated at points of the momentum space that are symmetric with respect to the initial momentum of the injected electrons.

scholarly journals The weak cosmic censorship conjecture may be violated

2020 ◽  
Author(s):  
Wen-Xiang Chen
Keyword(s):  
Black Hole ◽  
Wave Function ◽  
Boundary Conditions ◽  
Cosmic Censorship ◽  
Traditional Theory ◽  
Schwarzschild Black Hole ◽  
Entropy Reduction ◽  
Set Up ◽  
Coupled Wave ◽  
Cosmic Censorship Conjecture

According to traditional theory, the Schwarzschild black hole does not produce superradiation. If the boundary conditions are set up in advance, this possibility will be combined with the boson-coupled wave function in the Schwarzschild black hole, where the incident boson will have a mirrored mass, so even the Schwarzschild black hole can generate superradiation phenomena.Recently, an article of mine obtained interesting results about the Schwarzschild black hole can generate superradiation phenomena. The result contains some conclusions that violate the "no-hair theorem". We know that the phenomenon of black hole superradiation is a process of entropy reduction I found that the weak cosmic censorship conjecture may be violated.

scholarly journals Must be the wave function single-valued? Ring vs. periodic boundary conditions, spinors and double rotations

2022 ◽  
Author(s):  
Josep Planelles
Keyword(s):  
Wave Function ◽  
Angular Momentum ◽  
Boundary Conditions ◽  
Undergraduate Students ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Unitary Groups ◽  
Lecture Notes

This is a lecture notes for undergraduate students. We try to tackle the single valuedness of spatial and double valuedness of spin functions. Also, we adress the need of spinors to accommodate spin functions with some parallelism to the need of axial vectors (or antisymmetric traceless tensors) to accommodate angular momentum. Finally, we revisit the Dirac and Weyl tricks on the non-equivalence of a 2 pi and a 4 pi rotation related the topology of rotation and unitary groups.

N-soliton solution for a higher-order Chen–Lee–Liu equation with nonzero boundary conditions

2019 ◽  
Vol 34 (04) ◽  
pp. 2050054 ◽  
Author(s):  
Yi Zhao ◽  
Engui Fan
Keyword(s):  
Boundary Conditions ◽  
Soliton Solution ◽  
Hilbert Problem ◽  
Asymptotic Properties ◽  
Soliton Solutions ◽  
Higher Order ◽  
Third Order ◽  
Third Order Dispersion ◽  
Nonzero Boundary ◽  
Riemann Hilbert Problem

In this paper, the Riemann–Hilbert approach is applied to investigate a higher-order Chen–Lee–Liu equation with third-order dispersion and quintic nonlinearity terms. Based on the analytical, symmetric and asymptotic properties of eigenfunctions, a generalized Riemann–Hilbert problem associated with Chen–Lee–Liu equation with nonzero boundary conditions is constructed. Further, the [Formula: see text]-soliton solution is found by solving the generalized Riemann–Hilbert problem. As an illustrative example, two kinds of one-soliton solutions with different forms of parameters are obtained.

scholarly journals Quantum Cosmology of Quadratic f(R) Theories with a FRW Metric

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
V. Vázquez-Báez ◽  
C. Ramírez
Keyword(s):  
Wave Function ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Quantum Cosmology ◽  
Free Parameter ◽  
Equations Of Motion ◽  
Frw Metric ◽  
The Universe

We study the quantum cosmology of a quadratic fR theory with a FRW metric, via one of its equivalent Horndeski type actions, where the dynamic of the scalar field is induced. The classical equations of motion and the Wheeler-DeWitt equation, in their exact versions, are solved numerically. There is a free parameter in the action from which two cases follow: inflation + exit and inflation alone. The numerical solution of the Wheeler-DeWitt equation depends strongly on the boundary conditions, which can be chosen so that the resulting wave function of the universe is normalizable and consistent with Hermitian operators.

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