scholarly journals Contact Interactions in One-Dimensional Quantum Mechanics: a Family of Generalized б'-Potentials

2019 ◽  
Vol 64 (11) ◽  
pp. 1021 ◽  
Author(s):  
A. V. Zolotaryuk
Keyword(s):  
Parameter Space ◽  
Resonant Tunneling ◽  
Measure Zero ◽  
Contact Point ◽  
Delta Function ◽  
Point Interactions ◽  
Dimensional Parameter ◽  
One Dimensional ◽  
The Family ◽  
Delta Derivative

A “one-point” approximation is proposed to investigate the transmission of electrons through the extra thin heterostructures composed of two parallel plane layers. The typical example is the bilayer for which the squeezed potential profile is the derivative of Dirac’s delta function. The Schr¨odinger equation with this singular one-dimensional profile produces a family of contact (point) interactions each of which (called a “distributional” б′-potential) depends on the way of regularization. The discrepancies widely discussed so far in the literature regarding the family of delta derivative potentials are eliminated using a two-scale power-connecting parametrization of the bilayer potential that enables one to extend the family of distributional б′-potentials to a whole class of “generalized” б′-potentials. In a squeezed limit of the bilayer structure to zero thickness, the resonant tunneling through this structure is shown to occur in the form of sharp peaks located on the sets of Lebesgue’s measure zero (called resonance sets). A four-dimensional parameter space is introduced for the representation of these sets. The transmission on the complement sets in the parameter space is shown to be completely opaque.

INTRINSIC RESONANT TUNNELING PROPERTIES OF THE ONE-DIMENSIONAL SCHRÖDINGER OPERATOR WITH A DELTA DERIVATIVE POTENTIAL

2013 ◽  
Vol 28 (01) ◽  
pp. 1350203 ◽  
Author(s):  
A. V. ZOLOTARYUK ◽  
Y. ZOLOTARYUK
Keyword(s):  
Schrödinger Operator ◽  
Resonant Tunneling ◽  
Bound States ◽  
Delta Function ◽  
Schrodinger Operator ◽  
Range Limit ◽  
One Dimensional ◽  
The One ◽  
Adjoint Operators ◽  
Delta Derivative

Restricting ourselves to a simple rectangular approximation but using properly a two-scale regularization procedure, additional resonant tunneling properties of the one-dimensional Schrödinger operator with a delta derivative potential are established, which appear to be lost in the zero-range limit. These "intrinsic" properties are complementary to the main already proved result that different regularizations of Dirac's delta function produce different limiting self-adjoint operators. In particular, for a given regularizing sequence, a one-parameter family of connection condition matrices describing bound states is constructed. It is proposed to consider the convergence of transfer matrices when the potential strength constant is involved into the regularization process, resulting in an extension of resonance sets for the transmission across a δ′-barrier.

THE PARAMETER SPACE OF A QUADRATIC FAMILY OF POLYNOMIAL MAPS OF ℂ2

2008 ◽  
Vol 19 (05) ◽  
pp. 541-556
Author(s):  
ALINA ANDREI
Keyword(s):  
Lyapunov Exponents ◽  
Parameter Space ◽  
Mandelbrot Set ◽  
General Context ◽  
One Dimensional ◽  
Polynomial Maps ◽  
Regular Maps ◽  
The Family ◽  
The One ◽  
Polynomial Family

In this paper, we study the parameter space of the quadratic polynomial family fλ,μ(z, w) = (λz + w2, μw + z2), which exhibits interesting dynamics. Two distinct subsets of the parameter space are studied as appropriate analogs of the one-dimensional Mandelbrot set and some of their properties are proved by using Lyapunov exponents. In the more general context of holomorphic families of regular maps, we show that the sum of the Lyapunov exponents is a plurisubharmonic function of the parameter, and pluriharmonic on the set of expanding maps. Moreover, for the family fλ,μ, we prove that the sum of the Lyapunov exponents is continuous.

The bandcount increment scenario. II. Interior structures

2008 ◽  
Vol 464 (2097) ◽  
pp. 2247-2263 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Canonical Form ◽  
Parameter Space ◽  
Complex Interaction ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Self Similar ◽  
Bifurcation Structures

Bifurcation structures in the two-dimensional parameter spaces formed by chaotic attractors alone are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In this second part, we investigate fine substructures nested into the basic structures reported and explained in part I. It is demonstrated that the overall structure of the chaotic domain is caused by a complex interaction of bandcount increment, bandcount adding and bandcount doubling structures, whereby some of them are nested into each other ad infinitum leading to self-similar structures in the parameter space.

scholarly journals Multistability and Self-Similarity in the Parameter-Space of a Vibro-Impact System

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Silvio L. T. de Souza ◽  
Iberê L. Caldas ◽  
Ricardo L. Viana
Keyword(s):  
Parameter Space ◽  
Smooth Boundary ◽  
Basins Of Attraction ◽  
Self Similarity ◽  
Dimensional Parameter ◽  
One Dimensional ◽  
Impact System ◽  
Periodic Sets ◽  
Self Similar ◽  
Rigid Walls

The dynamics of a dissipative vibro-impact system called impact-pair is investigated. This system is similar to Fermi-Ulam accelerator model and consists of an oscillating one-dimensional box containing a point mass moving freely between successive inelastic collisions with the rigid walls of the box. In our numerical simulations, we observed multistable regimes, for which the corresponding basins of attraction present a quite complicated structure with smooth boundary. In addition, we characterize the system in a two-dimensional parameter space by using the largest Lyapunov exponents, identifying self-similar periodic sets.

BIFURCATIONS OF ATTRACTING CYCLES FROM TIME-DELAYED CHUA’S CIRCUIT

1995 ◽  
Vol 05 (03) ◽  
pp. 653-671 ◽  
Author(s):  
Yu. L. MAISTRENKO ◽  
V.L. MAISTRENKO ◽  
S.I. VIKUL ◽  
L.O. CHUA
Keyword(s):  
Parameter Space ◽  
Rotation Number ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Homoclinic Bifurcation ◽  
Dimensional Parameter ◽  
Tent Map ◽  
One Dimensional ◽  
Skew Tent Map ◽  
Border Collision Bifurcation

We study the bifurcations of attracting cycles for a three-segment (bimodal) piecewise-linear continuous one-dimensional map. Exact formulas for the regions of periodicity of any rational rotation number (Arnold’s tongues) are obtained in the associated three-dimensional parameter space. It is shown that the destruction of any Arnold’s tongue is a result of a border-collision bifurcation, and is followed by the appearance of a cycle of intervals with the same rotation number, whose dynamics is determined by a skew tent map. Finally, for the interval cycle the merging bifurcation corresponds to a homoclinic bifurcation of some point cycle.

scholarly journals Measure of Similarity between GMMs by Embedding of the Parameter Space That Preserves KL Divergence

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 957
Author(s):  
Branislav Popović ◽  
Lenka Cepova ◽  
Robert Cep ◽  
Marko Janev ◽  
Lidija Krstanović
Keyword(s):  
Computational Complexity ◽  
Parameter Space ◽  
Recognition Accuracy ◽  
Gaussian Mixture Models ◽  
Gaussian Mixture ◽  
Dimensional Manifold ◽  
Dimensional Parameter ◽  
Trade Off ◽  
Measure Of Similarity ◽  
Lower Dimensional

In this work, we deliver a novel measure of similarity between Gaussian mixture models (GMMs) by neighborhood preserving embedding (NPE) of the parameter space, that projects components of GMMs, which by our assumption lie close to lower dimensional manifold. By doing so, we obtain a transformation from the original high-dimensional parameter space, into a much lower-dimensional resulting parameter space. Therefore, resolving the distance between two GMMs is reduced to (taking the account of the corresponding weights) calculating the distance between sets of lower-dimensional Euclidean vectors. Much better trade-off between the recognition accuracy and the computational complexity is achieved in comparison to measures utilizing distances between Gaussian components evaluated in the original parameter space. The proposed measure is much more efficient in machine learning tasks that operate on large data sets, as in such tasks, the required number of overall Gaussian components is always large. Artificial, as well as real-world experiments are conducted, showing much better trade-off between recognition accuracy and computational complexity of the proposed measure, in comparison to all baseline measures of similarity between GMMs tested in this paper.

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

2007 ◽  
Vol 17 (09) ◽  
pp. 3071-3083 ◽  
Author(s):  
J. M. GONZÀLEZ-MIRANDA
Keyword(s):  
Phase Diagram ◽  
Bifurcation Diagram ◽  
Parameter Space ◽  
Neuron Model ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Rapid Responses ◽  
Bifurcation Structures

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

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