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Methods Of Modern Mathematical Physics

By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as ${\mathcal{P}\mathcal{T}}$, the true meaning and significance of the so-called charge operators $\mathcal{C}$ and the ${\mathcal{C}\mathcal{P}\mathcal{T}}$-inner products,...

Pseudo-Hermitian Representation of Quantum Mechanics

We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of this class of driven-diffusive systems—which includes exclusion processes—focusing on interesting physical properties, such as shocks and phase transitions. We then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix-product form. In addition to a gentle introduction to this matrix-product approach, how it works and how it relates to similar constructions that arise in other physical contexts, we present a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed. We also review a number of more advanced topics, including nonequilibrium free-energy functionals, the classification of exclusion processes involving multiple particle species, existence proofs of a matrix-product state for a given model and more complicated variants of the matrix-product state that allow various types of parallel dynamics to be handled. We conclude with a brief discussion of open problems for future research.

Nonequilibrium steady states of matrix-product form: a solver's guide

A quantum graph is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian. Such graphs have been studied recently in relation to some problems of mathematics, physics and chemistry. The paper has a survey nature and is devoted to the description of some basic notions concerning quantum graphs, including the boundary conditions, self-adjointness, quadratic forms, and relations between quantum and combinatorial graph models.

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Quantum graphs: I. Some basic structures

Singular perturbations of Schrodinger type operators are of interest in mathematics, e.g. to study spectral phenomena, and in applications of mathematics in various sciences, e.g. in physics, chemistry, biology, and in technology. They also often lead to models in quantum theory which are solvable in the sense that the spectral characteristics (eigenvalues, eigenfunctions, and scattering matrix) can be computed. Such models then allow us to grasp the essential features of interesting and complicated phenomena and serve as an orientation in handling more realistic situations.

In the last ten years two books have appeared on solvable models in quantum theory built using special singular perturbations of Schrodinger operators. The book by S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden
"Solvable Models in Quantum Mechanics" describes the models in rigorous mathematical terms. It gives a detailed analysis of perturbations of the Laplacian in R^d, d=1,2,3, by potentials with support on a discrete finite or infinite set of point sources (chosen in a deterministic, respectively, stochastic manner). Physically these operators describe the motion of a quantum mechanical particle moving under the action of a potential supported, e.g., by the points of a crystal lattice or a random solid. Such systems and models are also described in physical terms in the book by Yu.N.Demkov and V.N.Ostrovsky "Zero-range Potentials in Atomic Physics", which also contains a description of applications in other areas such as in optics and electromagnetism. Let us also remark that a translation of the book by S.Albeverio, F.Gesztesy, R.Hoegh-Krohn and H.Holden in Russian has been published with additional comments and literature. Since the appearance of these books several important new developments have taken place. It is the main aim of the present book to present some of these new developments in a unified formalism which also puts some of the basic results of the preceding books into a new light. The new developments concern in particular a systematic study of finite rank perturbations of (self--adjoint) operators (in particular differential operators), of generalized (singular) perturbations, of the corresponding scattering theory as well as infinite rank perturbations and multiple particles (many--body) problems in quantum theory. We also present the theory of point interaction Hamiltonians, as a particular case of a general theory of singular perturbations of differential operators.
This theory has received steadily increasing attention over the years also for its many applications in physics (solid state physics, nuclear physics), electromagnetism (antennas), and technology (metallurgy, nanophysics).

We hope this monograph can serve as a basis for orientation
in a rapidly developing area of analysis, mathematical physics and their applications.

Singular Perturbations of Differential Operators

Differential (and more general self-adjoint) operators involving singular interactions arise naturally in a range of topics such as, classical and quantum physics, chemistry, and electronics. This book presents a systematic mathematical study of these operators, with particular emphasis on spectral and scattering problems. Suitable for researchers in analysis or mathematical physics, this book could also be used as a text for an advanced course on the applications of analysis.

Singular perturbations of differential operators : solvable Schrödinger type operators

The inverse scattering problem on branching graphs is studied. The definition of the Schrodinger operator on such graphs is discussed. The operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices. It is shown that in general the scattering matrix does not determine the topology of the graph, the potentials on the edges and the boundary conditions uniquely.

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On the inverse scattering problem on branching graphs

http://staff.math.su.se/kurasov/PDFARCHIV/JMAA96.pdf

Distribution Theory for Discontinuous Test Functions and Differential Operators with Generalized Coefficients

The inverse spectral problem for the Laplace operator on a finite metric graph is investigated. It is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2. To prove the result, a trace formula connecting the spectrum of the Laplace operator with the set of periodic orbits for the metric graph is established.

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Pavel Kurasov

Papers

Singular Perturbations of Differential Operators

Singular perturbations of differential operators : solvable Schrödinger type operators

On the inverse scattering problem on branching graphs

Distribution Theory for Discontinuous Test Functions and Differential Operators with Generalized Coefficients

Inverse spectral problem for quantum graphs