EUDML

If ${v_{i}}$ and ${w_{j}}$ are two families of unitary bases for $C^{n}$ , and $θ$ is a fixed number, let $V^{n}$ and $W^{n}$ be subspaces of $C^{n}$ spanned by $[θ \cdot n]$ vectors in ${v_{i}}$ and ${w_{j}}$ respectively. We study the angle between $V^{n}$ and $W^{n}$ as $n$ goes to infinity. We show that when ${v_{i}}$ and ${w_{j}}$ arise in certain arithmetically defined families, the angles between $V^{n}$ and $W^{n}$ may either tend to $0$ or be bounded away from zero, depending on the behavior of an associated eigenvalue problem.

Non-Sunada graphs

Robert Brooks — 1999

Annales de l'institut Fourier

We consider the question of whether there is a converse to the Sunada Theorem in the context of $k$ -regular graphs. We give a weak converse to the Sunada Theorem, which gives a necessary and sufficient condition for two graphs to be isospectral in terms of a Sunada-like condition, and show by example that a strong converse does not hold.

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The first eigenvalue of a Riemann surface.

Compactness of isospectral sets

Typical surfaces and random graphs

Isospectral graphs and isospectral surfaces

On the Spectrum of Non-Compact Manifolds with Finite Volume.

A Relation Between Growth and the Spectrum of the Laplacian.

Injectivity radius and low eigenvalues of hyperbolic manifolds.

The bottom of the spectrum of a Riemannian covering.

On branched coverings of 3-manifolds which fiber over the circle.

The fundamental group and the spectrum of the Laplacian.

On the angles between certain arithmetically defined subspaces of $𝐂^{n}$

Non-Sunada graphs