Spectral theory of differential operators

The study of the spectrum of partial differential equations such as the Schrödinger equation is a classical topic in mathematical physics whose importance can be traced back at least as far as the works of mathematicians such as Hilbert, Courant, Weyl and Pólya, and physicists such as Lord Rayleigh, Lorentz and Sommerfeld around the turn of the 20th Century. It is a specialty of GFM, with our members focusing on qualitative and quantitative properties of operators such as Laplacians and Schrödinger operators, in various contexts such as Euclidean domains, Riemannian manifolds and quantum graphs.

The emphasis of our work is on the link between analytic properties of the spectrum, the eigenvalues and eigenfunctions of the operator, and geometric properties of the underlying domain or manifold, such as via isoperimetric inequalities, or the influence of the geometry on the behaviour of extremal sets for spectral quantities.

This analysis is complemented by the development of rigorous new methods for the study of shape optimisation problems in spectral theory, which have applications not only supporting our theoretical studies and in forming and testing new conjectures, but also in classical and new problems coming from physical applications such as musical acoustics, and civil and mechanical engineering.

Currently, our members are working primarily on the following topics:

  • Asymptotic behaviour of the spectrum
  • Numerical methods for shape optimisation, including in 3D and 4D
  • Shape optimisation and isoperimetric inequalities in spectral theory
  • Spectral analysis of quantum graphs
  • Spectral determinants
  • Spectral minimal partitions

Many of our seminars are held in hybrid or pure online form as webinars in Lisbon WADE, a joint webinar series run by three research centres in Lisbon including GFM.

Members involved in this area: Pêdra Andrade, Pedro Antunes, Pedro Freitas, James Kennedy, Isabel Salavessa

Current students working in this area: Hernani Calunga, Simão Eusébio, Miguel Santiago, Vinicius Santos

Some recent students and postdocs: Andrea Serio, Corentin Léna, Davide Buoso, Matthias Hofmann, Gianpaolo Piscitelli, Ophélie Rouby

Some recent key publications:

  • P. R. S. Antunes, R. D. Benguria, V. Lotoreichik and T. Ourmières-Bonafos, A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities, Comm. Math. Phys. 386 (2021), 781-818
    URL: https://doi.org/10.1007/s00220-021-03959-6
  • G. Berkolaiko, J. B. Kennedy, P. Kurasov and D. Mugnolo, Surgery methods for the spectral analysis of quantum graphs, Trans. Amer. Math. Soc. 372 (2019), 5153-5197
    URL: https://doi.org/10.1090/tran/7864
  • P. Freitas and J. B. Kennedy, Extremal domains and Pólya-type inequalities for the Robin Laplacian on rectangles and unions of rectangles, Int. Math. Res. Not. IMRN 2021, 13730–13782 (2021)
    URL: https://doi.org/10.1093/imrn/rnz204
  • P. Freitas and R. S. Laugesen, From Neumann to Steklov and beyond, via Robin: the Weinberger way, Amer. J. Math. 143 (2021), 969–994
    URL: https://doi.org/10.1353/ajm.2021.0024
  • P. Freitas and I. Salavessa, Families of non-tiling domains satisfying Pólya’s conjecture, J. Math. Phys. 64 (2023), 121503
    URL: https://doi.org/10.1063/5.0161050

Some recent review articles:

  • P. R. S. Antunes and E. Oudet, Numerical results for extremal problem for eigenvalues of the Laplacian, Chapter 11 in A. Henrot (ed.), Shape optimization and spectral theory, De Gruyter Open, Warsaw, 2017
    URL: https://doi.org/10.1515/9783110550887
  • D. Bucur, P. Freitas and J. Kennedy, The Robin problem, Chapter 4 in A. Henrot (ed.), Shape optimization and spectral theory, De Gruyter Open, Warsaw, 2017
    URL: https://doi.org/10.1515/9783110550887