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This book introduces and discusses the self-adjoint extension problem for symmetric operators on Hilbert space. It presents the classical von Neumann and Krein–Vishik–Birman extension schemes both in their modern form and from a historical perspective, and provides a detailed analysis of a range of applications beyond the standard pedagogical examples (the latter are indexed in a final appendix for the reader’s convenience). Self-adjointness of operators on Hilbert space representing quantum observables, in particular quantum Hamiltonians, is required to ensure real-valued energy levels, unitary evolution and, more generally, a self-consistent theory. Physical heuristics often produce candidate Hamiltonians that are only symmetric: their extension to suitably larger domains of self-adjointness, when possible, amounts to declaring additional physical states the operator must act on in order to have a consistent physics, and distinct self-adjoint extensions describe different physics. Realising observables self-adjointly is the first fundamental problem of quantum-mechanical modelling. The discussed applications concern models of topical relevance in modern mathematical physics currently receiving new or renewed interest, in particular from the point of view of classifying self-adjoint realisations of certain Hamiltonians and studying their spectral and scattering properties. The analysis also addresses intermediate technical questions such as characterising the corresponding operator closures and adjoints. Applications include hydrogenoid Hamiltonians, Dirac–Coulomb Hamiltonians, models of geometric quantum confinement and transmission on degenerate Riemannian manifolds of Grushin type, and models of few-body quantum particles with zero-range interaction. Graduate students and non-expert readers will benefit from a preliminary mathematical chapter collecting all the necessary pre-requisites on symmetric and self-adjoint operators on Hilbert space (including the spectral theorem), and from a further appendix presenting the emergence from physical principles of the requirement of self-adjointness for observables in quantum mechanics.

Mathematical analysis --- Mathematics --- Mathematical physics --- analyse (wiskunde) --- wiskunde --- fysica --- Mathematical physics. --- Mathematical analysis. --- Sistemes hamiltonians

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Sistemes dinàmics diferenciables --- Dinàmica diferencial --- Exponents de Lyapunov --- Fluxos (Sistemes dinàmics diferenciables) --- Sistemes dinàmics aleatoris --- Sistemes dinàmics complexos --- Sistemes dinàmics hiperbòlics --- Sistemes hamiltonians --- Teoria de la bifurcació --- Dinàmica topològica --- Differentiable dynamical systems --- Data processing. --- Mathematical models. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Dinàmica combinatòria

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Differentiable dynamical systems. --- Sistemes dinàmics diferenciables --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Dinàmica diferencial --- Dinàmica combinatòria --- Exponents de Lyapunov --- Fluxos (Sistemes dinàmics diferenciables) --- Sistemes dinàmics aleatoris --- Sistemes dinàmics complexos --- Sistemes dinàmics hiperbòlics --- Sistemes hamiltonians --- Teoria de la bifurcació --- Dinàmica topològica

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Explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition. In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics. This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience. It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology. Dynamical Systems in Neuroscience presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons. It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties. The book introduces dynamical systems, starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems. Each chapter proceeds from the simple to the complex, and provides sample problems at the end. The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians. Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines. Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum--or taught by math or physics department in a way that is suitable for students of biology. This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience. An additional chapter on synchronization, with more advanced material, can be found at the author's website, www.izhikevich.com.

Differentiable dynamical systems. --- Neurology. --- Neurons. --- Neurosciences. --- Models, Neurological --- Neurons --- Neurosciences --- Models, Neurological. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Nerve cells --- Neurocytes --- Neural sciences --- Neurological sciences --- Neuroscience --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Cells --- Nervous system --- Medicine --- Neuropsychiatry --- Medical sciences --- Diseases --- Differentiable dynamical systems --- Neurology --- NEUROSCIENCE/General --- Neurologia --- Electrofisiologia --- Sistemes dinàmics diferenciables --- Electricitat animal --- Electrobiologia --- Fisiologia --- Anestèsia elèctrica --- Circuit neuronal --- Electrocardiografia --- Electroencefalografia --- Músculs --- Potencials evocats (Electrofisiologia) --- Medicina --- Inhibició --- Neurocirurgia --- Neurooftalmologia --- Neuroimmunologia --- Neurologia pediàtrica --- Neurologia geriàtrica --- Neurologia veterinària --- Psicofisiologia --- Psicopatologia --- Urgències en neurologia --- Neuròlegs --- Dinàmica diferencial --- Exponents de Lyapunov --- Fluxos (Sistemes dinàmics diferenciables) --- Sistemes dinàmics aleatoris --- Sistemes dinàmics complexos --- Sistemes dinàmics hiperbòlics --- Sistemes hamiltonians --- Teoria de la bifurcació --- Dinàmica topològica --- Dinàmica combinatòria

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This book provides an accessible overview concerning the stochastic numerical methods inheriting long-time dynamical behaviours of finite and infinite-dimensional stochastic Hamiltonian systems. The long-time dynamical behaviours under study involve symplectic structure, invariants, ergodicity and invariant measure. The emphasis is placed on the systematic construction and the probabilistic superiority of stochastic symplectic methods, which preserve the geometric structure of the stochastic flow of stochastic Hamiltonian systems. The problems considered in this book are related to several fascinating research hotspots: numerical analysis, stochastic analysis, ergodic theory, stochastic ordinary and partial differential equations, and rough path theory. This book will appeal to researchers who are interested in these topics.

Numerical analysis. --- Dynamical systems. --- Probabilities. --- Numerical Analysis. --- Dynamical Systems. --- Probability Theory. --- Sistemes hamiltonians --- Equacions diferencials estocàstiques --- Grups simplèctics --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Mathematical analysis --- Grups algebraics lineals --- Anàlisi numèrica --- Equació de Fokker-Planck --- Mecànica de Hamilton --- Mecànica hamiltoniana --- Sistemes de Hamilton --- Sistemes dinàmics de Hamilton --- Sistemes dinàmics diferenciables --- Equacions de Hamilton-Jacobi --- Hamiltonian systems. --- Stochastic differential equations. --- Symplectic groups. --- Groups, Symplectic --- Linear algebraic groups --- Differential equations --- Fokker-Planck equation --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Differentiable dynamical systems