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

The five sections of these notes will one day be the first five chapters of a book, to appear some time after 2001.

Spectra and pseudospectra

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 Schrodinger 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 Schrodinger 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

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Mathematical Methods in Quantum Mechanics

Introduction Uniform estimates on the $p$-Laplacian, limits as $p\to\infty$ The transport set and transport rays Differentiability and smoothness properties of the potential Generic properties of transport rays Behavior of the transport density along rays Vanishing of the transport density at the ends of rays Approximate mass transfer plans Passage to limits a.e. Optimality Appendix: Approximating semiconcave and semiconvex functions by $C^2$ functions Bibliography.

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Differential equations methods for the Monge-Kantorovich mass transfer problem

Reviews - Eigenfunction Expansions Associated with Second-Order Differential Equations. By E. C. Titchmarsh Pp. 175. 20s. 1946. (Oxford University)

Hardy-Sobolev-type inequalities associated with the operator L:={x}⋅∇ are established, using an improvement to the Sobolev embedding theorem obtained by M.Ledoux. The analysis involves the determination of the operator semigroup {e −tL ∗ L } 0≠t≥0.

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On inequalities of hardy–sobolev type

The repeated diagonalization techniques used by Eastham to obtain asymptotic results for solutions to linear differential systems are further developed to produce a numerical algorithm for estimating the Titchmarsh-Weyl m -coefficient in the second-order case. These analytic results have been exploited to produce a computer code (RDML1) to generate these solutions and to obtain precise global error bounds.

Repeated diagonalization and the numerical computation of the Titchmarsh-Weyl m(λ)-function

In 1979 Copson proved the following analogue of the Hardy-Littlewood inequality: if is a sequence of real numbers such that are convergent, where Δa n = a n+1 – a n and Δ 2 a n = Δ(Δa n ), then is convergent and the constant 4 being best possible. Equality occurs if and only if a n = 0 for all n. In this paper we give a result that extends Copson's result to inequalities of the form where Mx n =–Δ(p n_l Δx n_l )+q n x n (n = 0, 1, …). The validity of such an inequality and the best possible value of the constant K are determined in terms of the analogue of the Titchmarsh-Weyl m -function for the difference equation Mx n = λw n x n (n = 0, 1, …).

William Desmond Evans

Papers

On inequalities of hardy–sobolev type

Repeated diagonalization and the numerical computation of the Titchmarsh-Weyl m(λ)-function

On an extension of Copson's inequality for infinite series

Interval Limit-Point Criteria for Differential Expressions and their Powers

On the Deficiency Indices of Powers of Real 2nth-Order Symmetric Differential Expressions