Quasi-Normal Modes, Non-Selfadjoint Operators and Pseudospectrum: an Interdisciplinary Approach

José Luis Jaramillo^1*Piotr Bizon²Edgar Gasperín³

• ¹Universite Bourgogne Europe, Dijon, France
• ²Uniwersytet Jagiellonski w Krakowie, Kraków, Poland
• ³Universidade de Lisboa, Lisbon, Portugal

The notion of normal mode pervades physics, providing a common conceptual and technical thread among different subfields of research and a bedrock for the study of (conservative) linear dynamics. The key mathematical property underlying normal modes is the diagonalisability of selfadjoint operators in terms of an orthonormal basis, guaranteed by the "spectral theorem". The validity of the later extends to operators commuting with their adjoint, namely "normal operators". Normal modes and their associated spectrum ("normal frequencies") stand as a cornerstone of the dynamics driven by such normal operators.A fundamental change occurs in non-conservative systems driven by non-selfadjoint, more generally, non-normal operators. Familiar normal modes are then substituted by "quasi-normal modes" (QNM), that encode invariantly the characteristic linear response to perturbations and indeed share some of the features of normal modes. However, the loss of the spectral theorem critically impact QNMs: their completeness is not guaranteed, their orthogonality is lost and the corresponding eigenvalues in the spectrum are potentially unstable under small perturbations. "Non-normal dynamics" Trefethen and Embree (2005) driven by these operators are then subject to characteristic non-normal effects absent in the normal case, such as spectral instabilities in QNM frequencies, growth transients or pseudo-resonances Trefethen et al. (1993). These differences ultimately respond to a key contrast between the normal and non-normal theory: the respective structural status of the spectrum. Whereas the spectrum of normal time-generator operators provides a tight control of the full dynamics, in the non-normal case such control is not guaranteed by the spectrum alone -except for late dynamics-and specific tools from non-selfadjoint spectral theory are required.Non-modal analysis (e.g. Schmid (2007)) provides a framework for the study of non-normal dynamics, crucially incorporating concepts and tools from the (spectral) theory of non-selfadjoint operators. This Research Topic highlights the notion of pseudospectrum and its relation to QNMs and its properties.Pseudospectra sets in the complex frequency plane contain in particular the (QNM) spectrum set, but encode more information crucial to seize all the dynamics, in particular the above-mentioned non-normal effects. However, whereas the spectrum concept is built only on the operator itself, the pseudospectrum does depend on the choice of scalar product and associated norm. The same applies for other key non-modal analysis tools, in particular the "growth function", crucial in the study of non-modal growth transients and the assessment of optimal disturbances(see below). The question of the choice of scalar product becomes a central theme in non-modal analysis of non-normal dynamics, and in particular in this Research Topic.We present now the articles in this Research Topic. Given the interrelation among the contributions, the adopted arrangement is somewhat "ad hoc", but we hope it illustrates the interdisciplinarity of the subject, both inside the physics community and in its connection with ongoing mathematical developments.1. Quasi-normal modes as an eigenvalue problem.A key prerequisite for the application of non-normal dynamical concepts to scattering and QNMS is to cast the dynamics in terms of non-selfadjoint (non-normal) time generator. Building on the spacetime geometrical insights developed in general relativity, Panosso Macedo and Zenginoglu reviews the so-called hyperboloidal approach to scattering on black hole (BH) spacetimes. This geometric scheme provides the required non-selfadjoint operator, non-normality being associated to losses through spacetime boundaries. BH QNMs are then cast as eigenvalues of a non-selfadjoint spectral problem. Applications to QNM excitation factors, spectral QNM instability and quadratic QNMs are presented. An interesting extension of such hyperboloidal scheme to problems with dispersion (e.g. some quantum-gravity motivated problems) is presented in Burgess and König.Another approach to cast resonances (QNMs) as an eigenvalue problem is the so-called complex scaling method (see e.g. Simon (1978) and references therein). Although such a method is referred to in Warnick. and Vogel, it is not really discussed in this Research Topic. However Richarte et al.discusses a spectral problem reminiscent of such complex scaling. They review the approach to QNMs as analytical continuations of bound states of an appropriate selfadjoint operator and, in particular, signal a non-selfadjoint spectral issue: the breakdown of the method for recovering QNMs whose analytic continuation is not in the domain where the operator defining the bound states is selfadjoint.2. Non-normal spectral and dynamical aspects in gravitational (black hole) physics.First indications of a non-normal behaviour of QNMs in gravity were implicit in the seminal results on BH QNM instabilities presented in Aguirregabiria and Vishveshwara (1996); Vishveshwara (1996) and in Nollert (1996); Nollert and Price (1999). However, it was precisely the hyperboloidal approach reviewed in Panosso Macedo and Zenginoglu that allowed to establish transparently in Jaramillo et al.(2021) the essential role of non-normal phenomena. That was the starting point of a rapid development of non-normal dynamics in the gravitational context, one of the main subject in this Research Topic. Warnick presents the first mathematically rigorous account in the literature of the BH QNM instability phenomenon. Characterised as a "perturbative" spectral instability, an analytical discussion in terms of QNM "modes" and "co-modes" is presented. Sensitivity to perturbations is Bizo ń et al. enhanced as damping increases, a feature explained in terms of the need to control higher derivatives to properly define increasing QNM overtones. This feature can be (dually) interpreted as a spectral stability in high-order Sobolev norms or, equivalently, as an spectral instability in the "energy norm", consequence of distributing energy in small scales. A generalisation of pseudospectra is introduced, tailored to the non-normality of operators appearing in the BH scattering problem.In a complementary work, Boyanov disentangles the confusion between the instability discussed in Warnick from another important BH QNM instability referred to as the "flea on the elephant" (reminiscent of Simon (1985)). These two instabilities respond to distinct instability mechanisms, the first one corresponding to the "perturbation instability" of (already) existing QNMs, whereas the second one involves a "branch instability" with the appearance of a new family of QNMs.Other aspects of pseudospectra and spectral instability are presented in the articles Areán et al Wu and Lalanne reviews the remarkable development that QNM theory has recently experienced in optics and plasmonics. Focus is placed on the role of QNM theory for designing and understanding micro and nano-resonators, that play a key role in current photonics, with an emphasis on the notions of "mode hybridization" and "mode perturbation". As an instance of optics-gravity interdisciplinarity, and motivated by the study of the fiber optical soliton, Burgess and König presents an adaptation of the spacetime hyperboloidal approach to scattering and QNMs (see also Al Sheikh ( 2022)) to settings with dispersion. This is crucial in optics and in some dispersive modified gravity theories.5. Mathematical aspects of non-normality.We collect here the articles in the Research Topic covering mathematical subjects and ranging from pseudospectra, spectral instability and random perturbations, oriented graph and networks or partial differential equation (PDE) hyperbolicity. They harmonically complement the physical contributions: 5.i. Krejčiřík and Siegl discusses the pseudospectrum from the perspective of "pseudomodes", a notion of approximate eigenvector not to be confused with QNMs, although relevant in the spectral instability of the latter. In particular, a construction of pseudomodes for large "pseudoeigenvalues" is Frontiers