The Spectral Flow of Families of Self-Adjoint Operators

1993 ◽  
pp. 138-160
Author(s):  
Bernhelm Booß-Bavnbek ◽  
Krzysztof P. Wojciechowski
Keyword(s):  
Spectral Flow ◽  
Adjoint Operators

scholarly journals The KO-valued spectral flow for skew-adjoint Fredholm operators

2020 ◽  
pp. 1-52
Author(s):  
Chris Bourne ◽  
Alan L. Carey ◽  
Matthias Lesch ◽  
Adam Rennie
Keyword(s):  
Real Hilbert Space ◽  
Axiomatic Characterization ◽  
Comprehensive Treatment ◽  
Spectral Flow ◽  
Type Result ◽  
Fredholm Operators ◽  
Clifford Index ◽  
Clifford Module ◽  
Adjoint Operators ◽  
Topological Phases Of Matter

In this paper, we give a comprehensive treatment of a “Clifford module flow” along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in [Formula: see text] via the Clifford index of Atiyah–Bott–Shapiro. We develop its properties for both bounded and unbounded skew-adjoint operators including an axiomatic characterization. Our constructions and approach are motivated by the principle that [Formula: see text] That is, we show how the KO-valued spectral flow relates to a KO-valued index by proving a Robbin–Salamon type result. The Kasparov product is also used to establish a [Formula: see text] result at the level of bivariant K-theory. We explain how our results incorporate previous applications of [Formula: see text]-valued spectral flow in the study of topological phases of matter.

Localization of the spectrum of certain non-self-adjoint operators

1968 ◽  
Vol 3 (4) ◽  
pp. 264-266
Author(s):  
M. M. Gekhtman
Keyword(s):  
Adjoint Operators

GROUND STATE CHARGE OF FERMION SOLITON SYSTEM IN 1+1 AND 3+1 DIMENSIONS

1993 ◽  
Vol 08 (04) ◽  
pp. 705-721
Author(s):  
M. RAVENDRANADHAN ◽  
M. SABIR
Keyword(s):  
Ground State ◽  
Energy State ◽  
Flow Analysis ◽  
Background Field ◽  
Bound State ◽  
Fermion Mass ◽  
Spectral Flow ◽  
Bound State Spectrum ◽  
Spectral Asymmetry ◽  
State Charge

Ground state charge of some fermion soliton system without C-invariance is calculated in 1+1 and 3+1 dimensions by a combination of adiabatic method and spectral flow analysis. Induced charge is calculated by evolving adiabatically the fields from a vacuum having a background field which has a zero energy state and spectral symmetry. The spectral flow is calculated by an analysis of the bound state spectrum. In 1+1 dimension our calculations are in agreement with the results already found in the literature. In 3+1 dimension we study the interaction of fermions with monopoles and dyons. In the case of monopoles, even though there is spectral asymmetry, ground state charge is found to be ±1/2. It is shown that ground state charge gets contribution only from the lowest angular momentum states and is discontinuous at the fermion mass.

scholarly journals Singular Perturbations of Self-Adjoint Operators

2003 ◽  
Vol 6 (4) ◽  
pp. 349-384 ◽  
Author(s):  
Vladimir Derkach ◽  
Seppo Hassi ◽  
Henk de Snoo
Keyword(s):  
Singular Perturbations ◽  
Adjoint Operators

scholarly journals Higher spectral flow and an entire bivariant JLO cocycle

2012 ◽  
Vol 11 (1) ◽  
pp. 183-232 ◽  
Author(s):  
Moulay-Tahar Benameur ◽  
Alan L. Carey
Keyword(s):  
Dirac Operator ◽  
Chern Character ◽  
Dirac Operators ◽  
Cyclic Cohomology ◽  
Closed Manifold ◽  
Spectral Flow ◽  
Smooth Functions ◽  
Base Manifold ◽  
K Theory ◽  
Higher Spectral Flow

AbstractFor a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

scholarly journals Functions of perturbed tuples of self-adjoint operators

2012 ◽  
Vol 350 (7-8) ◽  
pp. 349-354 ◽  
Author(s):  
Fedor Nazarov ◽  
Vladimir Peller
Keyword(s):  
Adjoint Operators
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