scholarly journals Dimensional Reduction and Scattering Formulation for Even Topological Invariants

2020 ◽  
Author(s):  
Hermann Schulz-Baldes ◽  
Daniele Toniolo
Keyword(s):  
Hilbert Space ◽  
Half Space ◽  
Fermi Energy ◽  
Dimensional Reduction ◽  
Invertible Operator ◽  
Topological Invariants ◽  
Exponential Map ◽  
Cayley Transform ◽  
Reflection Matrix ◽  
Set Up

AbstractStrong invariants of even-dimensional topological insulators of independent Fermions are expressed in terms of an invertible operator on the Hilbert space over the boundary. It is given by the Cayley transform of the boundary restriction of the half-space resolvent. This dimensional reduction is routed in new representation for the K-theoretic exponential map. It allows to express the invariants via the reflection matrix at the Fermi energy, for the scattering set-up of a wire coupled to the half-space insulator.

Representation of Symmetries and Observables in Functional Hilbert Space. II.

1972 ◽  
Vol 27 (1) ◽  
pp. 7-22 ◽  
Author(s):  
A. Rieckers
Keyword(s):  
Hilbert Space ◽  
Relativistic Quantum ◽  
Preceding Paper ◽  
Complex Hilbert Space ◽  
Unbounded Operators ◽  
Test Function ◽  
Test Function Space ◽  
Quantum Observables ◽  
Set Up ◽  
Real Test

Abstract The representation of infinitesimal generators corresponding to the group representation dis-cussed in the preceding paper is analyzed in the Hilbert space of functionals over real test functions. Explicit expressions for these unbounded operators are constructed by means of the functio-nal derivative and by canonical operator pairs on dense domains. The behaviour under certain basis transformations is investigated, also for non-Hermitian generators. For the Hermitian ones a common, dense domain is set up where they are essentially selfadjoint. After having established a one-to-one correspondence between the real test function space and a complex Hilbert space the theory of quantum observables is applied to the functional version of a relativistic quantum field theory.

scholarly journals Some Properties of D-Operator on Hilbert Space

2020 ◽  
pp. 3366-3371
Author(s):  
Eiman Al-janabi
Keyword(s):  
Hilbert Space ◽  
Direct Product ◽  
Hilbert Spaces ◽  
Scalar Product ◽  
Invertible Operator ◽  
D Operator ◽  
New Type ◽  
Equivalent Property ◽  
Drazin Invertible Operator

In this paper, we introduce a new type of Drazin invertible operator on Hilbert spaces, which is called D-operator. Then, some properties of the class of D-operators are studied. We prove that the D-operator preserves the scalar product, the unitary equivalent property, the product and sum of two D-operators are not D-operator in general but the direct product and tenser product is also D-operator.

scholarly journals From quantum link models to D-theory: a resource efficient framework for the quantum simulation and computation of gauge theories

2021 ◽  
Vol 380 (2216) ◽  
Author(s):  
Uwe-Jens Wiese
Keyword(s):  
Hilbert Space ◽  
Gauge Theory ◽  
Dimensional Reduction ◽  
Gauge Theories ◽  
Quantum Simulation ◽  
Quantum Spin ◽  
Coulomb Phase ◽  
Finite Dimensional ◽  
Link Model ◽  
Quantum Link

Quantum link models provide an extension of Wilson’s lattice gauge theory in which the link Hilbert space is finite-dimensional and corresponds to a representation of an embedding algebra. In contrast to Wilson’s parallel transporters, quantum links are intrinsically quantum degrees of freedom. In D-theory, these discrete variables undergo dimensional reduction, thus giving rise to asymptotically free theories. In this way ( 1 + 1 ) -d C P ( N − 1 ) models emerge by dimensional reduction from ( 2 + 1 ) -d S U ( N ) quantum spin ladders, the ( 2 + 1 ) -d confining U ( 1 ) gauge theory emerges from the Abelian Coulomb phase of a ( 3 + 1 ) -d quantum link model, and ( 3 + 1 ) -d QCD arises from a non-Abelian Coulomb phase of a ( 4 + 1 ) -d S U ( 3 ) quantum link model, with chiral quarks arising naturally as domain wall fermions. Thanks to their finite-dimensional Hilbert space and their economical mechanism of reaching the continuum limit by dimensional reduction, quantum link models provide a resource efficient framework for the quantum simulation and computation of gauge theories. This article is part of the theme issue ‘Quantum technologies in particle physics’.

Genericity of Certain Classes of Unitary and Self-Adjoint Operators

1998 ◽  
Vol 41 (2) ◽  
pp. 137-139 ◽  
Author(s):  
J. R. Choksi ◽  
M. G. Nadkarni
Keyword(s):  
Hilbert Space ◽  
Conference Proceedings ◽  
Separable Hilbert Space ◽  
Strong Operator Topology ◽  
Slight Modification ◽  
Simple Spectrum ◽  
Cayley Transform ◽  
Unitary Operators ◽  
Full Support ◽  
Adjoint Operators

AbstractIn a paper [1], published in 1990, in a (somewhat inaccessible) conference proceedings, the authors had shown that for the unitary operators on a separable Hilbert space, endowed with the strong operator topology, those with singular, continuous, simple spectrum, with full support, forma dense Gδ. A similar theoremfor bounded self-adjoint operators with a given normbound (omitting simplicity) was recently given by Barry Simon [2], [3], with a totally different proof. In this note we show that a slight modification of our argument, combined with the Cayley transform, gives a proof of Simon’s result, with simplicity of the spectrum added.

SOME INEQUALITIES FOR THE NUMERICAL RADIUS FOR HILBERT SPACE OPERATORS

2016 ◽  
Vol 94 (3) ◽  
pp. 489-496 ◽  
Author(s):  
MOHSEN SHAH HOSSEINI ◽  
MOHSEN ERFANIAN OMIDVAR
Keyword(s):  
Hilbert Space ◽  
Invertible Operator ◽  
Numerical Radius ◽  
Hilbert Space Operators

We introduce some new refinements of numerical radius inequalities for Hilbert space invertible operators. More precisely, we prove that if $T\in {\mathcal{B}}({\mathcal{H}})$ is an invertible operator, then $\Vert T\Vert \leq \sqrt{2}\unicode[STIX]{x1D714}(T)$.

Dynamic Responses of Pile Groups Embedded in a Layered Poroelastic Half-Space to Harmonic Axial Loads

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Bin Xu ◽  
Jian-Fei Lu ◽  
Jian-Hua Wang
Keyword(s):  
Half Space ◽  
Middle Layer ◽  
Pile Group ◽  
Dynamic Responses ◽  
Fredholm Integral Equations ◽  
Pile Groups ◽  
Axial Loads ◽  
Reflection Matrix ◽  
Patch Load ◽  
Layered Half Space

The dynamic responses of a pile group embedded in a layered poroelastic half-space subjected to axial harmonic loads is investigated in this study. Based on Biot’s theory, the frequency domain fundamental solution for a vertical circular patch load applied in the layered poroelastic half-space is derived via the transmission and reflection matrix (TRM) method. Utilizing Muki’s method, the second kind of Fredholm integral equations describing the dynamic interaction between the layered half-space and the pile group is constructed. The proposed methodology was validated by comparing the results of this paper with a known result. Numerical results show that in a two-layered half-space, for the closely populated pile group with a rigid cap, the upper softer layer thickness has different influences on the central pile and the corner piles, while for the sparse pile group, it has almost the same influence on all the piles. For a three-layer half-space, the presence of a stiffer middle layer in the layered half-space will enhance the impedance of the pile group significantly, while a softer middle layer will reduce the impedance of the pile group.

Experimental investigation of PL modes in a single layer

1962 ◽  
Vol 52 (1) ◽  
pp. 59-66
Author(s):  
Freeman Gilbert ◽  
Stanley J. Laster
Keyword(s):  
Half Space ◽  
Arrival Time ◽  
Single Layer ◽  
Good Resolution ◽  
P Waves ◽  
Small Phase ◽  
Shear Modes ◽  
Phase And Group Velocities ◽  
Set Up ◽  
Group Velocities

Abstract A two dimensional seismic model has been set up to simulate the problem of elastic wave propagation in a single layer overlying a uniform half space. Both the source and the receiver are mounted on the free surface of the layer. Seismograms are presented as a funciton of range. In addition to the Rayleigh and shear modes, PL modes are observed. Experimentally determined phase and group velocities compare fairly well with theoretical curves. The decay factor for PL is maximum at the arrival time of P waves in the half-space. There is also a secondary maximum at the arrival time of P waves in the layer. Although the decay of PL is small, phase equalization of PL does not yield the initial pulse shape because the mode embraces an insufficient frequency band to permit good resolution.

Multiplicative Commutators of Operators

1966 ◽  
Vol 18 ◽  
pp. 737-749 ◽  
Author(s):  
Arlen Brown ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Invertible Operator ◽  
Analogous Question ◽  
Normal Operators

An invertible operator T on a Hilbert space is a multiplicative commutator if there exist invertible operators A and B on such that T = ABA–1B–1. In this paper we discuss the question of which operators are, and which are not, multiplicative commutators. The analogous question for additive commutators (operators of the form AB — BA) has received considerable attention and has, in fact, been completely settled (2). The present results represent the information we have been able to obtain by carrying over to the multiplicative problem the techniques that proved efficacious in the additive situation. While these results remain incomplete, they suffice, for example, to enable us to determine precisely which normal operators are multiplicative commutators.

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