Operator and Spectral Theory, University of Reading, UK, April 4 2018

General interests

  • Operator theory and complex analysis.
  • Concrete operators: Hankel, Toeplitz, singular integrals, integral operators, Carleson embeddings, etc.
  • Spectral theory and mathematical physics
  • Potential theory
  • Interplay between these interests and analytic number theory.

Specific interests

Classical analysis and multiplicative structure

I am interested in operators with multiplicative structure. For example, the multiplicative Hilbert matrix

\displaystyle \mathbf{H} = \left\{\frac{1}{\sqrt{nm}\log(nm)} \right\}_{n,m=2}^\infty,

considered as an operator on the sequence space \ell^2(\mathbb{N}_2). This is an example of a Helson matrix \{\alpha(nm)\}_{n,m}, also known as a multiplicative Hankel matrix.

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Spectra of double layer potentials 


Fig. 1: Generic L^2-spectrum for a single 3D-drop

Consider a metal nanoparticle, understood as a dielectric inclusion in infinite space. Depending on the material it’s made of, size and shape, it will exhibit different surface plasmon resonances, by which certain frequencies of light will be absorbed and scattered. This is related to remarkable optical phenomena of nanoparticles. For example, the intense colors which arise when silver or gold nanoparticles of particular shapes are suspended in a solution (Youtube-demonstration due to A. Orbaek White).

The surface plasmon resonances of a particle can be computed from its polarizability tensor, obtained by solving a set of electrostatic problems. From the point of view of spectral theory, the electrostatic problems can be associated with a spectral measure from which the polarizability tensor can be directly reconstructed. Points in the spectrum correspond to complex permittivities for which surface plasmon resonances can be excited.

I study the spectra of such problems using boundary integral equations, which leads to studying the spectra of double layer potential operators, in the case of the Laplacian also known as the Neumann-Poincaré operator.

When the particle’s surface has singular features, such as corners or edges, the spectrum ceases to be discrete; containing in addition to eigenvalues a continuum of points. Fig. 2 depicts a typical drop shape of opening angle 5\pi/18. Fig. 1 depicts the corresponding (essential) spectrum for square-integrable boundary data, the negative integers indicating the Fredholm index in the region which they appear. Each curve corresponds to a Fourier mode in the direction of rotational symmetry. Fig. 3 depicts the spectrum of the first mode for a drop with a large angle, for boundary data in the energy space. Note that the spectrum is real for this kind of boundary data, observing the physical relevance of solutions of finite energy.


Fig. 2: A 3D drop


Fig. 3: Energy spectrum of a 3D-drop

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