Published works
2025
Reflection positivity and its relation to disc, half plane and the strip,
Expositiones Mathematicae, doi.org/10.1016/j.exmath.2025.125660
(with Karl-Hermann Neeb and Maria Stella Adamo)
Abstract: We present a novel perspective on reflection positivity on the strip by systematically developing the analogies with the unit disc and the upper half plane in the complex plane. These domains correspond to the three conjugacy classes of one-parameter groups in the Möbius group (elliptic for the disc, parabolic for the upper half plane and hyperbolic for the strip). In all cases, reflection positive functions correspond to positive functionals on H^∞ for a suitable involution. For the strip, reflection positivity naturally connects with Kubo–Martin–Schwinger (KMS) conditions on the real line and further to standard pairs, as they appear in Algebraic Quantum Field Theory. We also exhibit a curious connection between Hilbert spaces on the strip and the upper half plane, based on a periodization process.
2023
Regular one-parameter groups, reflection positivity and their application to Hankel operators and standard subspaces,
PhD thesis at Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany, open.fau.de/handle/openfau/23744
(Advisor: Karl-Hermann Neeb)
Abstract: Standard subspaces are a well-studied object in algebraic quantum field theory (AQFT). Given a standard subspace V of a Hilbert space H, one is interested in unitary one-parameter groups on H with U_t V ⊆ V for every t ∈ R_+. If (V,U) is a non-degenerate standard pair on H, i.e. the self-adjoint infinitesimal generator of U is a positive operator with trivial kernel, two classical results are given by Borchers' Theorem, relating non-degenerate standard pairs to positive energy representations of the affine group Aff(R) and the Longo-Witten Theorem, stating the the semigroup of unitary endomorphisms of V can be identified with the semigroup of symmetric operator-valued inner functions on the upper half-plane. In this thesis, we prove results similar to the theorems of Borchers and of Longo-Witten for a more general framework of unitary one-parameter groups without the assumption that their infinitesimal generator is positive. We replace this assumption by the weaker assumption that the triple (H,V,U) is a so-called real regular one-parameter group.
2022
Reflection positivity and Hankel operators – The multiplicity free case,
Journal of Functional Analysis, doi.org/10.1016/j.jfa.2022.109493
(with Karl-Hermann Neeb and Maria Stella Adamo)
Abstract: We analyze reflection positive representations in terms of positive Hankel operators. This is motivated by the fact that positive Hankel operators are described in terms of their Carleson measures, whereas the compatibility condition between representations and reflection positive Hilbert spaces is quite intricate. This leads us to the concept of a Hankel positive representation of triples (G,S,τ ), where G is a group, τ an involutive automorphism of G and S ⊆ G a subsemigroup with τ(S) = S^−1 . For the triples (Z,N,−id_Z), corresponding to reflection positive operators, and (R,R_+,−id_R), corresponding to reflection positive one-parameter groups, we show that every Hankel positive representation can be made reflection positive by a slight change of the scalar product. A key method consists in using the measure μ_H on R_+ defined by a positive Hankel operator H on H²(C_+) to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for H.
2019
Chern numbers as half-signature of the spectral localizer,
Journal of Mathematical Physics, doi.org/10.1063/1.5094300
(with Hermann Schulz-Baldes and Edgar Lozano Viesca)
Abstract: Two recent papers proved that complex index pairings can be calculated as the half-signature of a finite dimensional matrix, called the spectral localizer. This paper contains a new proof of this connection for even index pairings based on a spectral flow argument. It also provides a numerical study of the spectral gap and the half-signature of the spectral localizer for a typical two-dimensional disordered topological insulator in the regime of a mobility gap at the Fermi energy. This regime is not covered by the above mathematical results (which suppose a bulk gap), but nevertheless the half-signature of the spectral localizer is a clear indicator of a topological phase.
Preprints
Outgoing monotone geodesics of standard subspaces
Abstract: We prove a real version of the Lax-Phillips Theorem and classify outgoing reflection positive orthogonal one-parameter groups. Using these results, we provide a normal form for outgoing monotone geodesics in the set Stand(H) of standard subspaces on some complex Hilbert space H. As the modular operators of a standard subspace are closely related to positive Hankel operators, our results are obtained by constructing some explicit symbols for positive Hankel operators. We also describe which of the monotone geodesics in Stand(H) arise from the unitary one-parameter groups described in Borchers' Theorem and provide explicit examples of monotone geodesics that are not of this type.