scholarly journals RT-Symmetric Laplace Operators on Star Graphs: Real Spectrum and Self-Adjointness

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Maria Astudillo ◽  
Pavel Kurasov ◽  
Muhammad Usman
Keyword(s):  
Quantum Mechanics ◽  
Spectral Properties ◽  
Block Structure ◽  
Circulant Matrices ◽  
Real Spectrum ◽  
Star Graph ◽  
Quantum Graphs ◽  
Star Graphs ◽  
Rotationally Symmetric ◽  
Symmetric Quantum

How ideas ofPT-symmetric quantum mechanics can be applied to quantum graphs is analyzed, in particular to the star graph. The class of rotationally symmetric vertex conditions is analyzed. It is shown that all such conditions can effectively be described by circulant matrices: real in the case of odd number of edges and complex having particular block structure in the even case. Spectral properties of the corresponding operators are discussed.

scholarly journals EIGENVALUES AND EIGENFUNCTIONS OF WOODS–SAXON POTENTIAL IN PT-SYMMETRIC QUANTUM MECHANICS

2006 ◽  
Vol 21 (27) ◽  
pp. 2087-2097 ◽  
Author(s):  
AYŞE BERKDEMIR ◽  
CÜNEYT BERKDEMIR ◽  
RAMAZAN SEVER
Keyword(s):  
Quantum Mechanics ◽  
Real Spectrum ◽  
Saxon Potential ◽  
Eigenvalues And Eigenfunctions ◽  
Discrete Energy ◽  
Energy Eigenvalues ◽  
Symmetric Quantum ◽  
Symmetric Version ◽  
Uvarov Method ◽  
S States

Using the Nikiforov–Uvarov method which is based on solving the second-order differential equations, we firstly analyzed the energy spectra and eigenfunctions of the Woods–Saxon potential. In the framework of the PT-symmetric quantum mechanics, we secondly solved the time-independent Schrödinger equation for the PT and non-PT-symmetric version of the potential. It is shown that the discrete energy eigenvalues of the non-PT-symmetric potential consist of the real and imaginary parts, but the PT-symmetric one has a real spectrum. Results are obtained for s-states only.

scholarly journals ${\mathcal{CPT}}$-CONSERVING HAMILTONIANS AND THEIR NONLINEAR SUPERSYMMETRIZATION USING DIFFERENTIAL CHARGE-OPERATORS ${\mathcal C}$

2005 ◽  
Vol 20 (30) ◽  
pp. 7107-7128 ◽  
Author(s):  
BIJAN BAGCHI ◽  
A. BANERJEE ◽  
EMANUELA CALICETI ◽  
FRANCESCO CANNATA ◽  
HENDRIK B. GEYER ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Differential Operator ◽  
Higher Order ◽  
Real Spectrum ◽  
New Class ◽  
Operator Form ◽  
Supersymmetric Version ◽  
Order Case ◽  
Recent Developments ◽  
Symmetric Quantum

A brief overview is given of recent developments and fresh ideas at the intersection of [Formula: see text]- and/or [Formula: see text]-symmetric quantum mechanics with supersymmetric quantum mechanics (SUSY QM). Within the framework of the resulting supersymmetric version of [Formula: see text]-symmetric quantum mechanics we study the consequences of the assumption that the "charge" operator [Formula: see text] is represented in a differential-operator form of the second or higher order. Besides the freedom allowed by the Hermiticity constraint for the operator [Formula: see text], encouraging results are obtained in the second-order case. In particular, the integrability of intertwining relations proves to match the closure of our nonlinear (viz., polynomial) SUSY algebra. In a particular illustration, our form of [Formula: see text]-symmetric SUSY QM leads to a new class of non-Hermitian polynomial oscillators with real spectrum which turn out to be [Formula: see text]-asymmetric.

NON-HERMITIAN QUANTUM FIELD THEORY

2005 ◽  
Vol 20 (19) ◽  
pp. 4646-4652 ◽  
Author(s):  
CARL. M. BENDER
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Degrees Of Freedom ◽  
Real Spectrum ◽  
Quantum Field ◽  
Unitary Theory ◽  
Symmetric Quantum ◽  
Quantum Mechanical Systems ◽  
Bounded Below

In my talk at the Seventh QCD Workshop held in Villefranche in January 2003 I showed that a non-Hermitian Hamiltonian H possessing an unbroken [Formula: see text] symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator [Formula: see text], which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate [Formula: see text] is cumbersome in quantum mechanics and impossible in quantum field theory. I describe here an alternative method for calculating [Formula: see text] directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method gives the [Formula: see text] operator in quantum field theory. The [Formula: see text] operator is a new time-independent observable in [Formula: see text]-symmetric quantum field theory.

Diagnosability of Bubble-Sort Star Graphs with Missing Edges

2019 ◽  
Vol 19 (02) ◽  
pp. 1950002 ◽  
Author(s):  
SHIYING WANG ◽  
YINGYING WANG
Keyword(s):  
Multiprocessor System ◽  
Star Graph ◽  
Star Graphs ◽  
Comparison Model ◽  
Model Following ◽  
Strong Property

The diagnosability of a multiprocessor system plays an important role. The bubble-sort star graph BSn has many good properties. In this paper, we study the diagnosis on BSn under the comparison model. Following the concept of the local diagnosability, the strong local diagnosability property is discussed. This property describes the equivalence of the local diagnosability of a node and its degree. We prove that BSn (n ≥ 5) has this property, and it keeps this strong property even if there exist (2n − 5) missing edges in it, and the result is optimal with respect to the number of missing edges.

Gyroscopically Stabilized Systems: A Class Of Quadratic Eigenvalue Problems With Real Spectrum

1991 ◽  
Vol 44 (1) ◽  
pp. 42-53 ◽  
Author(s):  
Lawrence Barkwell ◽  
Peter Lancaster ◽  
Alexander S. Markus
Keyword(s):  
Hilbert Space ◽  
Real Line ◽  
Spectral Properties ◽  
Eigenvalue Problems ◽  
Real Spectrum ◽  
Hyperbolic Operator ◽  
Quadratic Operator ◽  
Distribution Of Eigenvalues ◽  
Quadratic Eigenvalue Problems ◽  
Quadratic Eigenvalue

AbstractEigenvalue problems for selfadjoint quadratic operator polynomials L(λ) = Iλ2 + Bλ+ C on a Hilbert space H are considered where B, C∈ℒ(H), C >0, and |B| ≥ kI + k-l C for some k >0. It is shown that the spectrum of L(λ) is real. The distribution of eigenvalues on the real line and other spectral properties are also discussed. The arguments rely on the well-known theory of (weakly) hyperbolic operator polynomials.

scholarly journals On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 185
Author(s):  
Ram Band ◽  
Sven Gnutzmann ◽  
August Krueger
Keyword(s):  
Existence Of Solutions ◽  
Stationary Waves ◽  
Star Graph ◽  
Oscillation Theorem ◽  
Nodal Domains ◽  
Nodal Structure ◽  
Star Graphs ◽  
Spectral Curves ◽  
Nonlinear Quantum ◽  
Cubic Nonlinear Schrödinger Equation

We consider stationary waves on nonlinear quantum star graphs, i.e., solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e., the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting, such as the classic Sturm oscillation theorem.

scholarly journals Generation of families of spectra in PT -symmetric quantum mechanics and scalar bosonic field theory

2013 ◽  
Vol 371 (1989) ◽  
pp. 20120049 ◽  
Author(s):  
Steffen Schmidt ◽  
S. P. Klevansky
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Complex Analysis ◽  
One Dimension ◽  
Quantum Mechanical ◽  
Bosonic Field ◽  
Symmetric Quantum ◽  
The Difference ◽  
Do So

This paper explains the systematics of the generation of families of spectra for the -symmetric quantum-mechanical Hamiltonians H = p 2 + x 2 (i x ) ϵ , H = p 2 +( x 2 ) δ and H = p 2 −( x 2 ) μ . In addition, it contrasts the results obtained with those found for a bosonic scalar field theory, in particular in one dimension, highlighting the similarities to and differences from the quantum-mechanical case. It is shown that the number of families of spectra can be deduced from the number of non-contiguous pairs of Stokes wedges that display symmetry. To do so, simple arguments that use the Wentzel–Kramers–Brillouin approximation are used, and these imply that the eigenvalues are real. However, definitive results are in most cases presently only obtainable numerically, and not all eigenvalues in each family may be real. Within the approximations used, it is illustrated that the difference between the quantum-mechanical and the field-theoretical cases lies in the number of accessible regions in which the eigenfunctions decay exponentially. This paper reviews and implements well-known techniques in complex analysis and -symmetric quantum theory.

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