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)

We define generalized Lorentz-Zygmund spaces and obtain interpolation theorems for quasilinear operators on such spaces, using weighted Hardy inequalities. In the limiting cases of interpolation, we discover certain scaling property of these spaces and use it to obtain fine interpolation theorems in which the source is a sum of spaces and the target is an intersection of spaces. This yields a considerable improvement of the known results which we demonstrate with examples. We prove sharpness of the interpolation theorems by showing that the constraints on parameters are necessary for the interpolation theorems.

Interpolation of Operators on Scales of Generalized Lorentz‐Zygmund Spaces

This paper is concerned with the extension and generalization of a classic integral inequality first considered by Hardy and Littlewood in 1932. The integral inequality is $\big[\int^b\_a (p|f'|^2+q|f|^2)\big]^2 \leqslant K \int^b\_a w|f|^2 \int^b\_a w|w^{-1}[-(pf')'+qf]|^2 (f\in\Delta)$, where K is a positive number, the coefficients p, q and w are real-valued and w > 0 on the interval [a, b), - $\infty$ < a < b $\leqslant \infty$, and p$^{-1}$, q, w $\in$ L$\_{loc}$ [a, b); $\Delta$ is the maximal linear manifold of functions f for which the two integrals on the right-hand side are both finite. The Hardy-Littlewood inequality is the special case a = 0, b = $\infty$, p = w = 1 and q = 0 on [0, $\infty$); here K = 4 and all the cases of equality can be identified. In this paper the results of Everitt in 1971, connecting the above inequality with the Titchmarsh-Weyl m-coefficient of the associated differential equation -(py')' + qy = $\lambda$wy on [a,b), are restated in an improved form and are shown to hold under the above minimal conditions on the coefficients p, q and w. It is proved that the best possible value for the number K satisfies 1 < K $\leqslant \infty$, and that within these bounds all values of K can be realized. In the course of establishing these results new information, of independent interest, is obtained about the behaviour of the m-coefficient near the real axis. Examples are given to exhibit the wide range of possibilities for the integral inequality. Recent results on the asymptotic form of the m-coefficient, and on the transformation theory of the associated differential equation are shown to have applications for the general properties of the inequality. All recent contributions to this general integral inequality are listed in the references and their content is surveyed during the course of the paper.

William Desmond Evans

Papers

Interpolation of Operators on Scales of Generalized Lorentz‐Zygmund Spaces

A Return to the Hardy-Littlewood Integral Inequality

Approximation Numbers of Certain Volterra Integral Operators

Weighted Hardy and Poincaré Inequalities on Trees

Two-sided estimates of the approximation numbers of certain Volterra integral operators