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)

The Weyl-Sims classification for a second-order ordinary differential equation with general complex coefficients is investigated. Connections are then established between the associated m-function and the spectral properties of maximal-accretive realisations of the differential expression.

On the spectrum of second order differential operators with complex coefficients

In this paper we consider extensions of positive operators. We study the connections between the von Neumann theory of extensions and characterisations of positive extensions via decompositions of the domain of the associated form. We apply the results to elliptic second order differential operators and look in particular at examples of the Laplacian on a disc and the Aharonov–Bohm operator.

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Positive Self-adjoint Operator Extensions with Applications to Differential Operators

In Edmunds et al. [J Lond Math Soc 78(2):65–84, 2008], a representation of a compact linear operator $T$ acting between reflexive Banach spaces $X$ and $Y$ with strictly convex duals was established in terms of elements $x_n \in X,$ projections $P_n$ of $X$ onto subspaces $X_n$ which are such that $ \cap X_n = \mathrm {ker}{\rm T},$ and linear projections $S_n$ satisfying $S_n x = \sum _{j=1}^{n-1} \xi _j(x) x_j,$ where the coefficients $\xi _j(x)$ are given explicitly. If $\mathrm {ker}{\rm T} = \{0\}$ and the condition 

$$\begin{aligned} (A): \quad sup \Vert S_n \Vert < \infty \end{aligned}$$

is satisfied, the representation reduces to an analogue of the Schmidt representation of $T$ when $X$ and $Y$ are Hilbert spaces, and also $(x_n)$ is a Schauder basis of $X$; thus condition (A) can not be satisfied if $X$ does not have the approximation property. In this paper we investigate circumstances in which (A) does or does not hold, and analyse the implications.

The structure of compact linear operators in Banach spaces

We study the embeddings E : W(X(Ω), Y(Ω)) ↪ Z(Ω), where X(Ω), Y(Ω) and Z(Ω) are rearrangement–invariant Banach function spaces (BFS) defined on a generalized ridged domain Ω, and W denotes a first–order Sobolev–type space. We obtain two–sided estimates for the measure of non–compactness of E when Z(Ω) = X(Ω) and, in turn, necessary and sufficient conditions for a Poincare–type inequality to be valid and also for E to be compact. The results are used to analyse the example of a trumpet–shaped domain Ω in Lorentz spaces. We consider the problem of determining the range of possible target spaces Z(Ω), in which case we prove that the problem is equivalent to an analogue on the generalized ridge Γ of Ω. The range of target spaces Z(Ω) is determined amongst a scale of (weighted) Lebesgue spaces for “rooms and passages” and trumpet–shaped domains.

William Desmond Evans

Papers

On the spectrum of second order differential operators with complex coefficients

Positive Self-adjoint Operator Extensions with Applications to Differential Operators

The structure of compact linear operators in Banach spaces

Ridged Domains, Embedding Theorems and Poincaré Inequalities

A Solution to the General Bessel Moment Problem