scholarly journals On Concrete Spectral Properties of a Twisted Laplacian Associated with a Central Extension of the Real Heisenberg Group

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Aymane El Fardi ◽  
Allal Ghanmi ◽  
Ahmed Intissar
Keyword(s):  
Heisenberg Group ◽  
Central Extension ◽  
Spectral Properties ◽  
The Real ◽  
Invariance Properties ◽  
Heisenberg Type ◽  
Twisted Laplacian

We consider the special magnetic Laplacian given byΔν,μ=4∑j=1n∂2/∂zj∂zj¯+2iνE+E¯+n+2μE-E¯-ν2+μ2z2. We show thatΔν,μis connected to the sub-Laplacian of a group of Heisenberg type given byC ×ω Cnrealized as a central extension of the real Heisenberg groupH2n+1. We also discuss invariance properties ofΔν,μand give some of their explicit spectral properties.

scholarly journals Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1060
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A. del del Olmo
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Heisenberg Group ◽  
Real Line ◽  
Hermite Functions ◽  
Unitary Representations ◽  
Weyl Groups ◽  
The Real ◽  
Symmetry Properties ◽  
The Fourier Transform

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Weyl–Heisenberg group and some of their extensions.

On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks

2008 ◽  
Vol 28 (2) ◽  
pp. 423-446 ◽  
Author(s):  
Y. GUIVARC’H ◽  
EMILE LE PAGE
Keyword(s):  
Random Walk ◽  
Spectral Properties ◽  
Markov Operator ◽  
Dominant Eigenvalue ◽  
Fractional Powers ◽  
Stable Law ◽  
Fourier Operator ◽  
Transfer Operators ◽  
Heisenberg Type ◽  
Normal Law

AbstractWe consider a random walk on the affine group of the real line, we denote by P the corresponding Markov operator on $\mathbb {R}$, and we study the Birkhoff sums associated with its trajectories. We show that, depending on the parameters of the random walk, the normalized Birkhoff sums converge in law to a stable law of exponent α∈ ]0,2[ or to a normal law. The corresponding analysis is based on the spectral properties of two families of associated transfer operators Pt,Tt. The operator Pt is a Fourier operator and is considered here as a perturbation of the Markov operator P=P0 of the random walk. The operator Tt is related to Pt by a symmetry of Heisenberg type and is also considered as a perturbation of the Markov operator T0=T. We prove that these operators have an isolated dominant eigenvalue which has an asymptotic expansion involving fractional powers of t. The parameters of this expansion have simple expressions in terms of tails and moments of the stationary measures of P and T.

scholarly journals On the real rank of $C^\ast$-algebras of nilpotent locally compact groups

2012 ◽  
Vol 110 (1) ◽  
pp. 99 ◽  
Author(s):  
Robert J. Archbold ◽  
Eberhard Kaniuth
Keyword(s):  
Compact Group ◽  
Heisenberg Group ◽  
Identity Element ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Rank ◽  
Connected Component ◽  
Locally Compact ◽  
The Real

If $G$ is an almost connected, nilpotent, locally compact group then the real rank of the $C^\ast$-algebra $C^\ast (G)$ is given by $\operatorname {RR} (C^\ast (G)) = \operatorname {rank} (G/[G,G]) = \operatorname {rank} (G_0/[G_0,G_0])$, where $G_0$ is the connected component of the identity element. In particular, for the continuous Heisenberg group $G_3$, $\operatorname {RR} C^\ast (G_3))=2$.

scholarly journals A Comparison of Two Equivalent Real Formulations for Complex-Valued Linear Systems Part 2: Results

2003 ◽  
Vol 1 (4) ◽  
Author(s):  
Abnita Munankarmy ◽  
Michael Heroux
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Complex Plane ◽  
Spectral Properties ◽  
Complex Problem ◽  
Linear Solver ◽  
The Real ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods ◽  
Complex Valued

Many iterative linear solver packages focus on real-valued systems and do not deal well with complex-valued systems, even though preconditioned iterative methods typically apply to both real and complex-valued linear systems. Instead, commonly available packages such as PETSc and Aztec tend to focus on the real-valued systems, while complex-valued systems are seen as a late addition. At the same time, by changing the complex problem into an equivalent real formulation (ERF), a real valued solver can be used. In this paper we consider two ERF’s that can be used to solve complex-valued linear systems. We investigate the spectral properties of each and show how each can be preconditioned to move eigenvalues in a cloud around the point (1,0) in the complex plane. Finally, we consider an interleaved formulation, combining each of the previously mentioned approaches, and show that the interleaved form achieves a better outcome than either separate ERF. [This article is the second part of a sequence of reports. See the December 2002 issue for Part 1—Editor.]

scholarly journals Effects of Boundary Conditions on the Modal and Spectral Properties of the Shaft

2020 ◽  
Vol 22 (1) ◽  
pp. 42-47
Author(s):  
Petr Hruby ◽  
Tomas Nahlik ◽  
Dana Smetanova
Keyword(s):  
Boundary Conditions ◽  
Spectral Properties ◽  
Rotating Shaft ◽  
The Real ◽  
Vibration Problem ◽  
Modal Properties ◽  
Fixed End

Influence of boundary conditions (i.e. mounting type of shafts ends) on spectral and modal properties is studied in this paper. Cases with joints at both ends and with joint at one end and fixed end are described in detail. The vibration problem of rotating shaft is generalized to problem of vibration of the shaft in the rotating plane. The problem is illustrated on testing models. The real mounting type of shafts is presented.

Spectral Properties of the Dirac Operator on the Real Line

2021 ◽  
Vol 57 (2) ◽  
pp. 139-147
Author(s):  
A. G. Baskakov ◽  
I. A. Krishtal ◽  
N. B. Uskova
Keyword(s):  
Dirac Operator ◽  
Real Line ◽  
Spectral Properties ◽  
The Real

scholarly journals On the Singular Spectrum for Adiabatic Quasiperiodic Schrödinger Operators

2010 ◽  
Vol 2010 ◽  
pp. 1-30 ◽  
Author(s):  
Magali Marx ◽  
Hatem Najar
Keyword(s):  
Lyapunov Exponent ◽  
Asymptotic Formula ◽  
Real Line ◽  
Spectral Properties ◽  
Schrödinger Operators ◽  
Adiabatic Limit ◽  
Schrodinger Operators ◽  
The Real ◽  
Singular Spectrum ◽  
Momentum Direction

We study spectral properties of a family of quasiperiodic Schrödinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curve has a real branch that is extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent and show that the spectrum is purely singular. This result was conjectured and proved in a particular case by Fedotov and Klopp (2005).

scholarly journals On One Method of Studying Spectral Properties of Non-selfadjoint Operators

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Maksim V. Kukushkin
Keyword(s):  
Spectral Properties ◽  
Root Vector ◽  
Vector System ◽  
Real Component ◽  
Von Neumann ◽  
The Real ◽  
Selfadjoint Operators ◽  
Neumann Class ◽  
Nonselfadjoint Operator

In this paper, we explore a certain class of Non-selfadjoint operators acting on a complex separable Hilbert space. We consider a perturbation of a nonselfadjoint operator by an operator that is also nonselfadjoint. Our consideration is based on known spectral properties of the real component of a nonselfadjoint compact operator. Using a technique of the sesquilinear forms theory, we establish the compactness property of the resolvent and obtain the asymptotic equivalence between the real component of the resolvent and the resolvent of the real component for some class of nonselfadjoint operators. We obtain a classification of nonselfadjoint operators in accordance with belonging their resolvent to the Schatten-von Neumann class and formulate a sufficient condition of completeness of the root vector system. Finally, we obtain an asymptotic formula for the eigenvalues.

31.—Spectral Manifolds for Constant Coefficient Elliptic Operators in Lp(Rn)

1975 ◽  
Vol 72 (4) ◽  
pp. 359-369 ◽  
Author(s):  
M. Thompson
Keyword(s):  
Differential Operator ◽  
Real Axis ◽  
Constant Coefficient ◽  
Spectral Properties ◽  
Real Zero ◽  
Partial Differential Operator ◽  
Partial Differential ◽  
The Real ◽  
Resolution Of The Identity ◽  
Norm Of The Resolvent

SynopsisThe principal results of this paper concern the spectral properties of the maximal realisation Pp in Lp(Rn) of a formally self adjoint constant coefficient strongly elliptic partial differential operator P(D), assumed to be homogeneous of order 2m, for 1 ≦ p ≦ ∞ and n ≧ 2. If we assume that , for 2n/n + l ≦ p ≦ 2n/n−1, together with certain assumptions on the associated real zero surfaces P(ξ) = λ, λ > 0, then σ(Pp) = σc(Pp) = [0, ∞). We obtain an estimate on the norm of the resolvent of Pp for points near the real axis, which allows us to establish the existence of a generalised resolution of the identity in the sense of Kocan.

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