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This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph. Evolution equations on metric graphs have attracted much attention as effective tools for the modeling of particle and wave dynamics in branched structures and networks. Since branched structures and networks appear in different areas of contemporary physics with many applications in electronics, biology, material science, and nanotechnology, the development of effective modeling tools is important for the many practical problems arising in these areas. The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue is a representative sample of the works devoted to the solutions of these and other problems.

quantum graphs --- ground states --- open sets converging to metric graphs --- norm convergence of operators --- NLD --- scaling limit --- standing waves --- bound states --- networks --- localized nonlinearity --- nonlinear Schrödinger equation --- metric graphs --- convergence of spectra --- sine-Gordon equation --- NLS --- star graph --- point interactions --- Laplacians --- nonrelativistic limit --- nonlinear wave equations --- quantum graph --- soliton --- nonlinear shallow water equations --- Kre?n formula --- breather --- non-linear Schrödinger equation --- Schrödinger equation --- nodal structure

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Describes the generalized Sturmian method, which offers a fresh approach to the calculation of atomic spectra. This work also discusses methods for automatic generation of symmetry-adapted basis sets. A final chapter also discusses application of the generalized Sturmian method to the calculation of molecular spectra.

Quantum theory --- Schrödinger equation. --- Atomic spectra. --- Atoms --- Spectrum, Atomic --- Nuclear spectroscopy --- Spectrum analysis --- Equation, Schrödinger --- Schrödinger wave equation --- Differential equations, Partial --- Particles (Nuclear physics) --- Wave mechanics --- WKB approximation --- Mathematics. --- Spectra --- Schrodinger equation.

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The Schrödinger equation is the master equation of quantum chemistry. The founders of quantum mechanics realised how this equation underpins essentially the whole of chemistry. However, they recognised that its exact application was much too complicated to be solvable at the time. More than two generations of researchers were left to work out how to achieve this ambitious goal for molecular systems of ever-increasing size. This book focuses on non-mainstream methods to solve the molecular electronic Schrödinger equation. Each method is based on a set of core ideas and this volume aims to expla

Schrödinger equation. --- Wave functions. --- Wave mechanics. --- Electrodynamics --- Matrix mechanics --- Mechanics --- Molecular dynamics --- Quantum statistics --- Quantum theory --- Waves --- Wave function --- Functions --- Wave mechanics --- Configuration space --- Equation, Schrödinger --- Schrödinger wave equation --- Differential equations, Partial --- Particles (Nuclear physics) --- WKB approximation --- Schrodinger equation --- Wave functions

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In recent years there have been important and far reaching developments in the study of nonlinear waves and a class of nonlinear wave equations which arise frequently in applications. The wide interest in this field comes from the understanding of special waves called 'solitons' and the associated development of a method of solution to a class of nonlinear wave equations termed the inverse scattering transform (IST). Before these developments, very little was known about the solutions to such 'soliton equations'. The IST technique applies to both continuous and discrete nonlinear Schrödinger equations of scalar and vector type. Also included is the IST for the Toda lattice and nonlinear ladder network, which are well-known discrete systems. This book, first published in 2003, presents the detailed mathematical analysis of the scattering theory; soliton solutions are obtained and soliton interactions, both scalar and vector, are analyzed. Much of the material is not available in the previously-published literature.

Inverse scattering transform --- Nonlinear theories --- Schrödinger equation --- 517.988 --- Equation, Schrödinger --- Schrödinger wave equation --- Differential equations, Partial --- Particles (Nuclear physics) --- Wave mechanics --- WKB approximation --- 517.988 Nonlinear functional analysis and approximation methods --- Nonlinear functional analysis and approximation methods --- Nonlinear problems --- Nonlinearity (Mathematics) --- Calculus --- Mathematical analysis --- Mathematical physics --- Scattering transform, Inverse --- Transform, Inverse scattering --- Scattering (Mathematics) --- Transformations (Mathematics) --- Schrödinger equation. --- Nonlinear theories. --- Inverse scattering transform. --- Schrodinger equation.

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Quantization of physical systems requires a correct definition of quantum-mechanical observables, such as the Hamiltonian, momentum, etc., as self-adjoint operators in appropriate Hilbert spaces and their spectral analysis.  Though a “naïve”  treatment exists for dealing with such problems, it is based on finite-dimensional algebra or even infinite-dimensional algebra with bounded operators, resulting in paradoxes and inaccuracies.   A proper treatment of these problems requires invoking certain nontrivial notions and theorems from functional analysis concerning the theory of unbounded self-adjoint operators and the theory of self-adjoint extensions of symmetric operators. Self-adjoint Extensions in Quantum Mechanics begins by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes of the naïve treatment.  The necessary mathematical background is then built by developing the theory of self-adjoint extensions.  Through examination of  various quantum-mechanical systems, the authors show how quantization problems associated with the correct definition of observables and their spectral analysis can be treated consistently for comparatively simple quantum-mechanical systems.  Systems that are examined include free particles on an interval, particles in a number of potential fields including delta-like potentials, the one-dimensional Calogero problem, the Aharonov–Bohm problem, and the relativistic Coulomb problem. This well-organized text is most suitable for graduate students and postgraduates interested in deepening their understanding of mathematical problems in quantum mechanics beyond the scope of those treated in standard textbooks.  The book may also serve as a useful resource for mathematicians and researchers in mathematical and theoretical physics.

Quantum theory. --- Selfadjoint operators. --- Selfadjoint operators --- Quantum theory --- Civil & Environmental Engineering --- Physics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Atomic Physics --- Operations Research --- Applied Physics --- Mathematics --- Schrödinger equation. --- Dirac equation. --- Equation, Schrödinger --- Schrödinger wave equation --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Mathematics. --- Operator theory. --- Applied mathematics. --- Engineering mathematics. --- Mathematical physics. --- Physics. --- Quantum physics. --- Mathematical Physics. --- Mathematical Methods in Physics. --- Operator Theory. --- Quantum Physics. --- Applications of Mathematics. --- Differential equations, Partial --- Quantum field theory --- Wave equation --- Particles (Nuclear physics) --- Wave mechanics --- WKB approximation --- Mechanics --- Thermodynamics --- Math --- Science --- Functional analysis --- Physical mathematics --- Engineering --- Engineering analysis --- Mathematical analysis --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Schrodinger equation.