The Selberg trace formula for Dirac operators

Bölte, Jens; Stiepan, Hans-Michael

doi:10.1063/1.2359578

Cited by 5 publications

(11 citation statements)

References 32 publications

(40 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The spectral action functional, for a Dirac operator D of a spectral triple, is defined (see [5]), for f ∈ S(R) an even rapidly decaying test function, as In the case of a compact Riemann surface X of genus g ≥ 2, with the constant negative curvature hyperbolic metric, the behavior of the spectral action functional (4.8) is similar to the behavior of the Laplacian spectral action discussed in the previous subsection. The effect on the Selberg trace formula of replacing the Laplacian ∆ by the Dirac Laplacian D 2 is discussed in [4]. When we adapt these results to the argument given in the previous subsection, we obtain the following result for the spectral action of a hyperbolic surface.…”

Section: 2mentioning

confidence: 69%

“…In the specific case where the Riemann surface X is defined over a number field, so that it admits a Belyi map f : X = H\H → C = ∆ p,q,r \H and a uniformization X = H\H by a finite index subgroup of a triangle group ∆ p,q,r , as in [8], the automorphic functions approach of [4] to the spectral decomposition of the Dirac operator can be made more explicit, using the results of [14] on automorphic forms for triangle groups. S D,f (Λ) = Λ 2 (g(X) − 1) R rf (r) coth(πr)dr…”

Section: 2mentioning

confidence: 99%

“…Selberg zeta function and the Spectral Action. An approach to the Selberg trace formula via the Selberg zeta function is described in [16], see also §VII of [4] for the Dirac case. For (s) > 1 and (σ) > 1, the trace formula applied to the test function…”

Section: 3mentioning

confidence: 99%

“…This construction generalizes the case of the N = 4 Adinkras described in the previous Section 2. 4, which corresponds to the self-dual graphs. In the rest of this section we describe a different construction that associates origami curves to Adinkras.…”

Section: -Coloredmentioning

confidence: 99%

See 3 more Smart Citations

Adinkras, Dessins, Origami, and Supersymmetry Spectral Triples

Marcolli

Zolman

2019

P-Adic Num Ultrametr Anal Appl

View full text Add to dashboard Cite

show abstract

Section: 2mentioning

confidence: 69%

Section: 2mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: -Coloredmentioning

confidence: 99%

See 2 more Smart Citations

Adinkras, Dessins, Origami, and Supersymmetry Spectral Triples

Marcolli

Zolman

2019

P-Adic Num Ultrametr Anal Appl

View full text Add to dashboard Cite

show abstract

“…The 1D Dirac equation is briefly discussed in Section 1.1 of [66]: it was later extended to the case of networks and thoroughly studied by Bolte and his coauthors [9,10], who also observed that it then takes on each edge the form…”

Section: The Dirac Equationmentioning

confidence: 99%

Linear hyperbolic systems on networks: well-posedness and qualitative properties

2021

View full text Add to dashboard Cite

We study hyperbolic systems of one - dimensional partial differential equations under general , possibly non-local boundary conditions. A large class of evolution equations, either on individual 1- dimensional intervals or on general networks , can be reformulated in our rather flexible formalism , which generalizes the classical technique of first - order reduction . We study forward and backward well - posedness ; furthermore , we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first - order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied . p, li { white-space: pre-wrap; }

show abstract

Semigroup Methods for Evolution Equations on Networks

Mugnolo

2014

Understanding Complex Systems

186

226

View full text Add to dashboard Cite

Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems-cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science.Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse "real-life" situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications.Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence.The three major book publication platforms of the Springer Complexity program are the monograph series "Understanding Complex Systems" focusing on the various applications of complexity, the "Springer Series in Synergetics", which is devoted to the quantitative theoretical and methodological foundations, and the "Springer Briefs in Complexity" which are concise and topical working reports, case-studies, surveys, essays and lecture notes of relevance to the field. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.Future scientific and technological developments in many fields will necessarily depend upon coming to grips with complex systems. Such systems are complex in both their composition -typically many different kinds of components interacting simultaneously and nonlinearly with each other and their environments on multiple levels -and in the rich diversity of behavior of which they are capable.The Springer Series in Understanding Complex Systems series (UCS) promotes new strategies and paradigms for understanding and realizing applications of complex systems research in a wide variety of fields and endeavors. UCS is explicitly transdisciplinary. It has three main goals: First, to elaborate the concepts, methods and tools of complex systems at all levels of description and in all scientific fields, especially newly emerging areas within the life, social, behavioral, economic, neuro-and cognitive sciences (and derivatives thereof); second, to encourage novel applications of these ideas in various fields of engineering and computation such as robotics, nano-technology...

show abstract

The Selberg trace formula for Dirac operators

Cited by 5 publications

References 32 publications

Adinkras, Dessins, Origami, and Supersymmetry Spectral Triples

Adinkras, Dessins, Origami, and Supersymmetry Spectral Triples

Linear hyperbolic systems on networks: well-posedness and qualitative properties

Semigroup Methods for Evolution Equations on Networks

Contact Info

Product

Resources

About