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

We prove that for Diophantine ! and almost every µ; the almost Mathieu operator, (H!;‚;µ“)(n )=“ (n +1 ) +“ (ni 1) +‚ cos 2…(!n+µ)“(n), exhibits localization for ‚> 2 and purely absolutely continuous spectrum for ‚< 2:

Metal-insulator transition for the almost Mathieu operator

We prove the conjecture (known as the "Ten Martini Problem" after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies.

https://annals.math.princeton.edu/wp-content/uploads/annals-v170-n1-p08-p.pdf

The Ten Martini Problem

The extension theory of Hermitian operators and the moment problem

We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues. The corresponding eigenfunctions are exponentially well localized. These results hold in arbitrary dimension and with probability one. In one dimension, we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder. Our techniques extend to other physical systems which exhibit localization phenomena, such as infinite systems of coupled harmonic oscillators, or random Schrodinger operators in the continuum.

/pdf/constructive-proof-of-localization-in-the-anderson-tight-23i8fgjcj9.pdf

Constructive proof of localization in the Anderson tight binding model

Expansions in eigenfunctions of selfadjoint operators

Spectral Methods in Infinite-Dimensional Analysis

Spaces of Generalized Functions General Selfadjoint and Normal Operators Some Classes of Operators Acting in a Space of Functions of Infinitely Many Variables.

I︠u︡. M. Berezanskiĭ

Papers

Expansions in eigenfunctions of selfadjoint operators

Spectral Methods in Infinite-Dimensional Analysis

Selfadjoint operators in spaces of functions of infinitely many variables