[PDF] Spectral Analysis On Graph Like Spaces eBook

Author : Olaf Post
Publisher : Springer Science & Business Media
Page : 444 pages
File Size : 18,24 MB
Release : 2012-01-06
Category : Mathematics
ISBN : 3642238394

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Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances. Since the underlying spaces in the thin radius limit change, and become singular in the limit, we develop new tools such as norm convergence of operators acting in different Hilbert spaces, an extension of the concept of boundary triples to partial differential operators, and an abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original problem of graph-like spaces, so that they can be applied in many other situations where the spaces are perturbed.

Author : Nobuaki Obata
Publisher : Springer
Page : 0 pages
File Size : 16,50 MB
Release : 2017-02-23
Category : Science
ISBN : 9789811035050

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This book is designed as a concise introduction to the recent achievements on spectral analysis of graphs or networks from the point of view of quantum (or non-commutative) probability theory. The main topics are spectral distributions of the adjacency matrices of finite or infinite graphs and their limit distributions for growing graphs. The main vehicle is quantum probability, an algebraic extension of the traditional probability theory, which provides a new framework for the analysis of adjacency matrices revealing their non-commutative nature. For example, the method of quantum decomposition makes it possible to study spectral distributions by means of interacting Fock spaces or equivalently by orthogonal polynomials. Various concepts of independence in quantum probability and corresponding central limit theorems are used for the asymptotic study of spectral distributions for product graphs.This book is written for researchers, teachers, and students interested in graph spectra, their (asymptotic) spectral distributions, and various ideas and methods on the basis of quantum probability. It is also useful for a quick introduction to quantum probability and for an analytic basis of orthogonal polynomials.

Author : Christophe Cheverry
Publisher : Springer Nature
Page : 258 pages
File Size : 36,26 MB
Release : 2021-05-06
Category : Mathematics
ISBN : 3030674622

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This textbook provides a graduate-level introduction to the spectral theory of linear operators on Banach and Hilbert spaces, guiding readers through key components of spectral theory and its applications in quantum physics. Based on their extensive teaching experience, the authors present topics in a progressive manner so that each chapter builds on the ones preceding. Researchers and students alike will also appreciate the exploration of more advanced applications and research perspectives presented near the end of the book. Beginning with a brief introduction to the relationship between spectral theory and quantum physics, the authors go on to explore unbounded operators, analyzing closed, adjoint, and self-adjoint operators. Next, the spectrum of a closed operator is defined and the fundamental properties of Fredholm operators are introduced. The authors then develop the Grushin method to execute the spectral analysis of compact operators. The chapters that follow are devoted to examining Hille-Yoshida and Stone theorems, the spectral analysis of self-adjoint operators, and trace-class and Hilbert-Schmidt operators. The final chapter opens the discussion to several selected applications. Throughout this textbook, detailed proofs are given, and the statements are illustrated by a number of well-chosen examples. At the end, an appendix about foundational functional analysis theorems is provided to help the uninitiated reader. A Guide to Spectral Theory: Applications and Exercises is intended for graduate students taking an introductory course in spectral theory or operator theory. A background in linear functional analysis and partial differential equations is assumed; basic knowledge of bounded linear operators is useful but not required. PhD students and researchers will also find this volume to be of interest, particularly the research directions provided in later chapters.

Author : Bilal Ahmad Rather
Publisher : A.K. Publications
Page : 0 pages
File Size : 39,41 MB
Release : 2022-12-22
Category : Mathematics
ISBN : 9783258974040

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Spectral graph theory (Algebraic graph theory) is the study of spectral properties of matrices associated to graphs. The spectral properties include the study of characteristic polynomial, eigenvalues and eigenvectors of matrices associated to graphs. This also includes the graphs associated to algebraic structures like groups, rings and vector spaces. The major source of research in spectral graph theory has been the study of relationship between the structural and spectral properties of graphs. Another source has research in mathematical chemistry (theoretical/quantum chemistry). One of the major problems in spectral graph theory lies in finding the spectrum of matrices associated to graphs completely or in terms of spectrum of simpler matrices associated with the structure of the graph. Another problem which is worth to mention is to characterise the extremal graphs among all the graphs or among a special class of graphs with respect to a given graph, like spectral radius, the second largest eigenvalue, the smallest eigenvalue, the second smallest eigenvalue, the graph energy and multiplicities of the eigenvalues that can be associated with the graph matrix. The main aim is to discuss the principal properties and structure of a graph from its eigenvalues. It has been observed that the eigenvalues of graphs are closely related to all graph parameters, linking one property to another. Spectral graph theory has a wide range of applications to other areas of mathematical science and to other areas of sciences which include Computer Science, Physics, Chemistry, Biology, Statistics, Engineering etc. The study of graph eigen- values has rich connections with many other areas of mathematics. An important development is the interaction between spectral graph theory and differential geometry. There is an interesting connection between spectral Riemannian geometry and spectral graph theory. Graph operations help in partitioning of the embedding space, maximising inter-cluster affinity and minimising inter-cluster proximity. Spectral graph theory plays a major role in deforming the embedding spaces in geometry. Graph spectra helps us in making conclusions that we cannot recognize the shapes of solids by their sounds. Algebraic spectral methods are also useful in studying the groups and the rings in a new light. This new developing field investigates the spectrum of graphs associated with the algebraic structures like groups and rings. The main motive to study these algebraic structures graphically using spectral analysis is to explore several properties of interest.

Author : Gregory Berkolaiko
Publisher : American Mathematical Soc.
Page : 291 pages
File Size : 12,76 MB
Release : 2013
Category : Mathematics
ISBN : 0821892118

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A ``quantum graph'' is a graph considered as a one-dimensional complex and equipped with a differential operator (``Hamiltonian''). Quantum graphs arise naturally as simplified models in mathematics, physics, chemistry, and engineering when one considers propagation of waves of various nature through a quasi-one-dimensional (e.g., ``meso-'' or ``nano-scale'') system that looks like a thin neighborhood of a graph. Works that currently would be classified as discussing quantum graphs have been appearing since at least the 1930s, and since then, quantum graphs techniques have been applied successfully in various areas of mathematical physics, mathematics in general and its applications. One can mention, for instance, dynamical systems theory, control theory, quantum chaos, Anderson localization, microelectronics, photonic crystals, physical chemistry, nano-sciences, superconductivity theory, etc. Quantum graphs present many non-trivial mathematical challenges, which makes them dear to a mathematician's heart. Work on quantum graphs has brought together tools and intuition coming from graph theory, combinatorics, mathematical physics, PDEs, and spectral theory. This book provides a comprehensive introduction to the topic, collecting the main notions and techniques. It also contains a survey of the current state of the quantum graph research and applications.

Author : Gilbert Helmberg
Publisher : Elsevier
Page : 362 pages
File Size : 18,32 MB
Release : 2014-11-28
Category : Science
ISBN : 1483164179

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North-Holland Series in Applied Mathematics and Mechanics, Volume 6: Introduction to Spectral Theory in Hilbert Space focuses on the mechanics, principles, and approaches involved in spectral theory in Hilbert space. The publication first elaborates on the concept and specific geometry of Hilbert space and bounded linear operators. Discussions focus on projection and adjoint operators, bilinear forms, bounded linear mappings, isomorphisms, orthogonal subspaces, base, subspaces, finite dimensional Euclidean space, and normed linear spaces. The text then takes a look at the general theory of linear operators and spectral analysis of compact linear operators, including spectral decomposition of a compact selfadjoint operator, weakly convergent sequences, spectrum of a compact linear operator, and eigenvalues of a linear operator. The manuscript ponders on the spectral analysis of bounded linear operators and unbounded selfadjoint operators. Topics include spectral decomposition of an unbounded selfadjoint operator and bounded normal operator, functions of a unitary operator, step functions of a bounded selfadjoint operator, polynomials in a bounded operator, and order relation for bounded selfadjoint operators. The publication is a valuable source of data for mathematicians and researchers interested in spectral theory in Hilbert space.