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In diesem Lehrbuch werden die mathematischen Grundlagen exakt und
dennoch anschaulich und gut nachvollziehbar vermittelt. Sie werden
durchgehend anhand zahlreicher Musterbeispiele illustriert, durch
Anwendungen in der Informatik motiviert und durch historische
Hintergrunde oder Ausblicke in angrenzende Themengebiete
aufgelockert. Am Ende jedes Kapitels befinden sich Kontrollfragen,
die das Verstandnis testen und typische Fehler bzw.
Missverstandnisse ausraumen. Zusatzlich helfen zahlreiche
Aufwarmubungen (mit vollstandigem Losungsweg) und weiterfuhrende
Ubungsaufgaben das Erlernte zu festigen und praxisrelevant
umzusetzen. Dieses Lehrbuch ist daher auch sehr gut zum
Selbststudium geeignet. Erganzend wird in eigenen Abschnitten das
Computeralgebrasystem Mathematica vorgestellt und eingesetzt,
wodurch der Lehrstoff visualisiert und somit das Verstandnis
erleichtert werden kann.
In diesem Lehrbuch werden die mathematischen Grundlagen exakt und
dennoch anschaulich und gut nachvollziehbar vermittelt. Sie werden
durchgehend anhand zahlreicher Musterbeispiele illustriert, durch
Anwendungen in der Informatik motiviert und durch historische
Hintergrunde oder Ausblicke in angrenzende Themengebiete
aufgelockert. Am Ende jedes Kapitels befinden sich Kontrollfragen,
die das Verstandnis testen und typische Fehler bzw.
Missverstandnisse ausraumen. Zusatzlich helfen zahlreiche
Aufwarmubungen (mit vollstandigem Loesungsweg) und weiterfuhrende
UEbungsaufgaben das Erlernte zu festigen und praxisrelevant
umzusetzen. Dieses Lehrbuch ist daher auch sehr gut zum
Selbststudium geeignet. Erganzend wird in eigenen Abschnitten das
Computeralgebrasystem Mathematica vorgestellt und eingesetzt,
wodurch der Lehrstoff visualisiert und somit das Verstandnis
erleichtert werden kann.
Quantum mechanics and the theory of operators on Hilbert space have
been deeply linked since their beginnings in the early twentieth
century. States of a quantum system correspond to certain elements
of the configuration space and observables correspond to certain
operators on the space. This book is a brief, but self-contained,
introduction to the mathematical methods of quantum mechanics, with
a view towards applications to Schroedinger operators. Part 1 of
the book is a concise introduction to the spectral theory of
unbounded operators. Only those topics that will be needed for
later applications are covered. The spectral theorem is a central
topic in this approach and is introduced at an early stage. Part 2
starts with the free Schroedinger equation and computes the free
resolvent and time evolution. Position, momentum, and angular
momentum are discussed via algebraic methods. Various mathematical
methods are developed, which are then used to compute the spectrum
of the hydrogen atom. Further topics include the nondegeneracy of
the ground state, spectra of atoms, and scattering theory. This
book serves as a self-contained introduction to spectral theory of
unbounded operators in Hilbert space with full proofs and minimal
prerequisites: Only a solid knowledge of advanced calculus and a
one-semester introduction to complex analysis are required. In
particular, no functional analysis and no Lebesgue integration
theory are assumed. It develops the mathematical tools necessary to
prove some key results in nonrelativistic quantum mechanics.
Mathematical Methods in Quantum Mechanics is intended for beginning
graduate students in both mathematics and physics and provides a
solid foundation for reading more advanced books and current
research literature. This new edition has additions and
improvements throughout the book to make the presentation more
student friendly.
This volume contains twenty contributions in the area of
mathematical physics where Fritz Gesztesy made profound
contributions. There are three survey papers in spectral theory,
differential equations, and mathematical physics, which highlight,
in particular, certain aspects of Gesztesy's work. The remaining
seventeen papers contain original research results in diverse areas
reflecting his interests. The topics of these papers range from
stochastic differential equations; operators on graphs; elliptic
partial differential equations; Sturm-Liouville, Jacobi, and CMV
operators; semigroups; to inverse problems.
As a partner to Volume 1: Dimensional Continuous Models, this
monograph provides a self-contained introduction to
algebro-geometric solutions of completely integrable, nonlinear,
partial differential-difference equations, also known as soliton
equations. The systems studied in this volume include the Toda
lattice hierarchy, the Kac-van Moerbeke hierarchy, and the
Ablowitz-Ladik hierarchy. An extensive treatment of the class of
algebro-geometric solutions in the stationary as well as
time-dependent contexts is provided. The theory presented includes
trace formulas, algebro-geometric initial value problems,
Baker-Akhiezer functions, and theta function representations of all
relevant quantities involved. The book uses basic techniques from
the theory of difference equations and spectral analysis, some
elements of algebraic geometry and especially, the theory of
compact Riemann surfaces. The presentation is constructive and
rigorous, with ample background material provided in various
appendices. Detailed notes for each chapter, together with an
exhaustive bibliography, enhance understanding of the main results.
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