Wolpert , Shing-Tung Yau. File: PDF, 2. Shing-Tung Yau Chief Editor. File: DJVU, Tomasz Mrowka , Shing-Tung Yau. File: PDF, 3. File: PDF, 1. Alexander Grigoryan , Shing-Tung Yau.

File: PDF, File: DJVU, 5. Si Li Editor , Bong H. Coates , Shing-Tung Yau ed. File: DJVU, 3. File: DJVU, 2. File: PDF, 9. Shing-tung Yau , Steve Nadi. File: MOBI , 2. ZAlerts allow you to be notified by email about the availability of new books according to your search query. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a Finite Element Method FEM one can achieve good accuracy for the scattering matrix.

For example, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible to identify arithmetic objects using FEM. PDF arXiv:math. Under some reasonable restrictions, we state localisation theorems for the pair-eigenvalues and discuss relations to a class of non-self-adjoint spectral problems.

For the standard metric the spectrum is known.

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Our aim is to analyse the behaviour of eigenvalues when the metric is perturbed in an arbitrary smooth fashion from the standard one. We derive explicit asymptotic formulae for the two eigenvalues closest to zero. Note that these eigenvalues remain double eigenvalues under perturbations of the metric: they cannot split because of a particular symmetry of the Dirac operator in dimension three it commutes with the antilinear operator of charge conjugation. Our asymptotic formulae show that in the first approximation our two eigenvalues maintain symmetry about zero and are completely determined by the increment of Riemannian volume.

Spectral asymmetry is observed only in the second approximation of the perturbation process. As an example we consider a special family of metrics, the so-called generalized Berger spheres, for which the eigenvalues can be evaluated explicitly. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian. Additionally, we relate the asymptotics of complex eigenvalues to the two-term asymptotics of the eigenvalues of associated self-adjoint operators. We prove that the spectrum is qualitatively different when a certain parameter c equals 0, and when it is non-zero, and that certain features of the spectrum depend on Diophantine properties of c.

It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue.

The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation a peculiar feature of dimension 3. We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly.

We also establish a relation between our asymptotic formula and the eta invariant. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained.

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In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential.

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## Surveys in Differential Geometry: Eigenvalues of Laplacians and Other - Google книги

We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented. We now establish Schatten class properties of the associated resolvent operator. The results presented are motivated by and extend those recently found by various authors Benilov et al.

Our algorithm is part analytical and part numerical and is essentially a combination of four classical approaches domain decomposition, boundary elements, finite elements and spectral methods each of which is used in its most natural context. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided estimates for this term in a variety of situations.

We use a recently developed technique, the so called quadratic projection method, in order to achieve convergence free from spectral pollution. Spectral gap. A note on the isoperimetric constant Peter Buser. Logarithmic Harnack Inequalities Fan R. Chung , Shuehlin Yau. Parabolic Harnack inequality for divergence form second order differential operators Laurent Saloff-Coste. Li , Shuehlin Yau. Parabolic Harnack inequality and estimates of Markov chains on graphs Thierry Delmotte.

Ricci curvature and eigenvalue estimate on locally finite graphs Yong Lin , Shuehlin Yau. On the measure contraction property of metric measure spaces.