scholarly journals Bound states in the continuum and time evolution of the generalized eigenfunctions

2018 ◽  
Vol 64 (5) ◽  
pp. 464 ◽  
Author(s):  
David Lohr ◽  
Enriqueta Hernandez ◽  
Antonio Jauregui ◽  
Alfonso Mondragon
Keyword(s):  
Time Evolution ◽  
Bound States ◽  
Scattering Problem ◽  
Bound State ◽  
The Other ◽  
Von Neumann ◽  
Generalized Eigenfunctions ◽  
Jost Solutions ◽  
Type Potential ◽  
The Continuum

We study the Jost solutions for the scattering problem of a von Neumann-Wigner type potential, constructed by means of a two times iterated and completely degenerated Darboux transformation. We show that for a particular energy the unnormalizedJost solutions coalesce to give rise to a Jordan cycle of rank two. Performing a pole decomposition of the normalized Jost solutions we find the generalized eigenfunctions: one is a normalizable function corresponding to the bound state in the continuum and the other is a bounded, non-normalizable function. We obtain the time evolution of these functions as pseudo-unitary, characteristic of a pseudo-Hermitian system.

scholarly journals Large Irredundant Sets in Operator Algebras

2019 ◽  
Vol 72 (4) ◽  
pp. 988-1023
Author(s):  
Clayton Suguio Hida ◽  
Piotr Koszmider
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Continuum Hypothesis ◽  
The Other ◽  
Open Problems ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
The Continuum

AbstractA subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.Other partial results and more open problems are presented.

scholarly journals (1+1)-Dirac bound states in one dimension, with position-dependent Fermi velocity and mass

Open Physics ◽  
2013 ◽  
Vol 11 (4) ◽  
Author(s):  
Omar Mustafa
Keyword(s):  
Harmonic Oscillator ◽  
Bound States ◽  
Bound State ◽  
One Dimension ◽  
Fermi Velocity ◽  
Dirac Particles ◽  
The Continuum ◽  
Oscillator Models

AbstractWe extend Panella and Roy’s [17] work for massless Dirac particles with position-dependent (PD) velocity. We consider Dirac particles where the mass and velocity are both position-dependent. Bound states in the continuum (BIC)-like and discrete bound state solutions are reported. It is observed that BIC-like solutions are not only feasible for the ultra-relativistic (massless) Dirac particles but also for Dirac particles with PDmass and PD-velocity that satisfy the condition m(x) v F2 (x) = A, where A ≥ 0 is constant. Dirac Pöschl-Teller and harmonic oscillator models are also reported.

scholarly journals Relativistic Friedrichs–Lee model and quark-pair creation model

2020 ◽  
Vol 80 (12) ◽  
Author(s):  
Zhi-Yong Zhou ◽  
Zhiguang Xiao
Keyword(s):  
Potential Model ◽  
Scattering Amplitude ◽  
Bound State ◽  
Pair Creation ◽  
The Other ◽  
Hadron Spectrum ◽  
Lee Model ◽  
Continuum States ◽  
Quark Potential ◽  
The Continuum

AbstractIn this paper, we present how the Friedrichs–Lee model could be extended to the relativistic scenario and be combined with the relativistic quark pair creation model in a consistent way. This scheme could be applied to study the “unquenched” effect of the meson spectra. As an example, if the lowest $$J^{PC}=0^{++}$$ J PC = 0 + + $$(u\bar{u}+d\bar{d})/\sqrt{2}$$ ( u u ¯ + d d ¯ ) / 2 bound state in the potential model is coupled to the $$\pi \pi $$ π π continuum, two resonance poles could be found from the scattering amplitude for the continuum states. One of them could correspond to the $$f_0(500)/\sigma $$ f 0 ( 500 ) / σ and the other probably $$f_0(1370)$$ f 0 ( 1370 ) . This scheme might shed more light on why extra states could appear in the hadron spectrum other than the prediction of the quark potential model.

scholarly journals Bound states in the continuum in resonant nanostructures: an overview of engineered materials for tailored applications

Nanophotonics ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shereena Joseph ◽  
Saurabh Pandey ◽  
Swagato Sarkar ◽  
Joby Joseph
Keyword(s):  
Bound States ◽  
Beam Steering ◽  
Functional Materials ◽  
Bound State ◽  
Dielectric Medium ◽  
Quality Factors ◽  
Theoretical Frameworks ◽  
Experimental Realization ◽  
Promising Tool ◽  
The Continuum

Abstract From theoretical model to experimental realization, the bound state in the continuum (BIC) is an emerging area of research interest in the last decade. In the initial years, well-established theoretical frameworks explained the underlying physics for optical BIC modes excited in various symmetrical configurations. Eventually, in the last couple of years, optical-BICs were exploited as a promising tool for experimental realization with advanced nanofabrication techniques for numerous breakthrough applications. Here, we present a review of the evolution of BIC modes in various symmetry and functioning mediums along with their application. More specifically, depending upon the nature of the interacting medium, the excitations of BIC modes are classified into the pure dielectric and lossy plasmonic BICs. The dielectric constituents are again classified as photonic crystal functioning in the subwavelength regime, influenced by the diffraction modes and metasurfaces for interactions far from the diffraction regime. More importantly, engineered functional materials evolved with the pure dielectric medium are explored for hybrid-quasi-BIC modes with huge-quality factors, exhibiting a promising approach to trigger the nanoscale phenomena more efficiently. Similarly, hybrid modes instigated by the photonic and plasmonic constituents can replace the high dissipative losses of metallic components, sustaining the high localization of field and high figure of merit. Further, the discussions are based on the applications of the localized BIC modes and high-quality quasi-BIC resonance traits in the nonlinear harmonic generation, refractometric sensing, imaging, lasing, nanocavities, low loss on-chip communication, and as a photodetector. The topology-controlled beam steering and, chiral sensing has also been briefly discussed.

scholarly journals Topological confinement of vortices in two-flavor dense QCD

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Yuki Fujimoto ◽  
Muneto Nitta
Keyword(s):  
Domain Wall ◽  
Quark Matter ◽  
Bound States ◽  
Bound State ◽  
The Other ◽  
Dense Quark Matter ◽  
Two Phases ◽  
Different Color ◽  
Superfluid Vortices ◽  
Aharonov Bohm

Abstract We find a novel confinement mechanism in the two-flavor dense quark matter proposed recently, that consists of the 2SC condensates and the P-wave diquark condensates of d-quarks. This quark matter exhibiting color superconductivity as well as superfluidity is classified into two phases; confined and deconfined phases of vortices. We establish that the criterion of the confinement is color neutrality of Aharonov-Bohm (AB) phases: vortices exhibiting color non-singlet AB phases are confined by the so-called AB defects to form color-singlet bound states. In the deconfined phase, the most stable vortices are non-Abelian Alice strings, which are superfluid vortices with fractional circulation and non-Abelian color magnetic fluxes therein, exhibiting color non-singlet AB phases. On the other hand, in the confined phase, these non-Abelian vortices are confined to either a baryonic or mesonic bound state in which constituent vortices are connected by AB defects. The baryonic bound state consists of three non-Abelian Alice strings with different color magnetic fluxes with the total flux canceled out connected by a domain wall junction, while the mesonic bound state consists of two non-Abelian Alice strings with the same color magnetic fluxes connected by a single domain wall. Interestingly, the latter contains a color magnetic flux in its core, but this can exist because of color neutrality of its AB phase.

Aharonov–Bohm effect in a Coulomb-type potential under the influence of a cutoff point

2021 ◽  
pp. 2150136
Author(s):  
K. Bakke
Keyword(s):  
Electric Fields ◽  
Bound States ◽  
Energy Levels ◽  
Bound State ◽  
Cutoff Point ◽  
Coulomb Type ◽  
Bohm Effect ◽  
Forbidden Region ◽  
Aharonov Bohm ◽  
Type Potential

We analyze the influence of a cutoff point on a Coulomb-type potential that stems from the interaction of an electron with electric fields. This cutoff point establishes a forbidden region for the electron. Then, we search for bound state solutions to the Schrödinger equation. In addition, we consider a rotating reference frame. We show that the effects of rotation break the degeneracy of the energy levels. Further, we discuss the Aharonov–Bohm effect for bound states.

scholarly journals Spectral Singularities do not Correspond to Bound States in the Continuum

10.14311/1813 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Ali Mostafazadeh
Keyword(s):  
Bound States ◽  
Von Neumann ◽  
Spectral Singularities ◽  
Scattering States ◽  
Made In ◽  
The Continuum

We show that, contrary to a claim made in arXiv:1011.0645, the von Neumann-Winger bound states that lie in the continuum of the scattering states are fundamentally different from Naimark’s spectral singularities.

scholarly journals Quasi-BIC laser enabled by high-contrast grating resonator for gas detection

Nanophotonics ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Haoran Zhang ◽  
Tao Wang ◽  
Jingyi Tian ◽  
Jiacheng Sun ◽  
Shaoxian Li ◽  
...  
Keyword(s):  
Bound States ◽  
Signal To Noise Ratio ◽  
Bound State ◽  
Gain Medium ◽  
Gas Detection ◽  
Visible Range ◽  
High Signal ◽  
High Contrast ◽  
High Contrast Grating ◽  
The Continuum

Abstract In this work, we propose and numerically investigate a two-dimensional microlaser based on the concept of bound states in the continuum (BIC). The device consists of a thin gain layer (Rhodamine 6G dye-doped silica) sandwiched between two high-contrast-grating layers. The structure supports various BIC modes upon a proper choice of topological parameters; in particular it supports a high-Q quasi-BIC mode when partially breaking a bound state in the continuum at Γ point. The optically-pumped gain medium provides sufficient optical gain to compensate the quasi-BIC mode losses, enabling lasing with ultra-low pump threshold (fluence of 17 μJ/cm2) and very narrow optical linewidth in the visible range. This innovative device displays distinguished sensing performance for gas detection, and the emission wavelength sensitively shifts to the longer wavelength with the changing of environment refractive index (in order of 5 × 10−4). The achieved bulk sensitivity is 221 nm/RIU with a high signal to noise ratio, and a record-high figure of merit reaches to 4420 RIU−1. This ultracompact and low threshold quasi-BIC laser facilitated by the ultra-narrow resonance can serve as formidable candidate for on-chip gas sensor.

scholarly journals Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem

2018 ◽  
Vol 474 (2213) ◽  
pp. 20180050 ◽  
Author(s):  
Anne-Sophie Bonnet-Ben Dhia ◽  
Lucas Chesnel ◽  
Vincent Pagneux
Keyword(s):  
Spectral Problem ◽  
Bound States ◽  
The Other ◽  
Trapped Modes ◽  
Perfectly Matched Layers ◽  
Complex Eigenvalues ◽  
Weak Reflection ◽  
Incident Waves ◽  
Matched Layers ◽  
The Continuum

We consider the reflection–transmission problem in a waveguide with obstacle. At certain frequencies, for some incident waves, intensity is perfectly transmitted and the reflected field decays exponentially at infinity. In this work, we show that such reflectionless modes can be characterized as eigenfunctions of an original non-self-adjoint spectral problem. To select ingoing waves on one side of the obstacle and outgoing waves on the other side, we use complex scalings (or perfectly matched layers) with imaginary parts of different signs. We prove that the real eigenvalues of the obtained spectrum correspond either to trapped modes (or bound states in the continuum) or to reflectionless modes. Interestingly, complex eigenvalues also contain useful information on weak reflection cases. When the geometry has certain symmetries, the new spectral problem enters the class of P T -symmetric problems.

scholarly journals Acousto-optic modulation of photonic bound state in the continuum

2020 ◽  
Vol 9 (1) ◽  
Author(s):  
Zejie Yu ◽  
Xiankai Sun
Keyword(s):  
Refractive Index ◽  
Integrated Circuits ◽  
Acoustic Waves ◽  
Bound States ◽  
Quantum Information Processing ◽  
Modulation Frequency ◽  
Surface Acoustic Waves ◽  
High Refractive Index ◽  
Bound State ◽  
The Continuum

AbstractPhotonic bound states in the continuum (BICs) have recently been studied in various systems and have found wide applications in sensors, lasers, and filters. Applying BICs in photonic integrated circuits enables low-loss light guidance and routing in low-refractive-index waveguides on high-refractive-index substrates, which opens a new avenue for integrated photonics with functional single-crystal materials. Here, we demonstrate high-quality integrated lithium niobate microcavities inside which the photonic BIC modes circulate and further modulate these BIC modes acousto-optically by using piezoelectrically actuated surface acoustic waves at microwave frequencies. With a high acousto-optic modulation frequency, the acousto-optic coupling is well situated in the resolved-sideband regime. This leads to coherent coupling between microwave and optical photons, which is exhibited by the observed electro-acousto-optically induced transparency and absorption. Therefore, our devices serve as a paradigm for manipulating and controlling photonic BICs on a chip, which will enable many other applications of photonic BICs in the areas of microwave photonics and quantum information processing.

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