Scalar quasinormal modes of Schwarzschild–anti-de Sitter black holes: spectral analysis and a formal coefficient deformation

Abstract

We study quasinormal modes (QNMs) of a minimally coupled massless scalar field on four-dimensional Schwarzschild–anti-de Sitter (SAdS) black holes using a Chebyshev spectral method. After compactifying the exterior domain, the radial problem is formulated as a quadratic matrix pencil in the dimensionless frequency. For the standard Dirichlet, or vanishing-field, boundary condition at the conformal AdS boundary, we reproduce the known scalar spectra across small, intermediate, and large black holes, including long overtone sequences and the expected approach to pure-AdS normal modes in the small black hole limit. We also impose a one-parameter relation between the slow and fast asymptotic coefficients. For a massless scalar in four dimensions, the slow branch lies outside the standard finite canonical energy domain. Because we neither construct a renormalized variational principle nor prove well-posedness of an evolution containing this branch, we present only the formal spectrum of the deformed radial problem. Each non-Dirichlet value examined yields an additional eigenvalue with positive imaginary part and hence, an exponentially growing separated formal mode. This does not establish a physical instability of the standard scalar theory. The interpolation parameter is representation-dependent. In a black-hole-independent conformal coordinate, the corresponding response coupling scales with the cube of the dimensionless horizon radius, so equal intermediate parameter values for different black-hole sizes do not represent the same boundary coupling.

Keywords

Bibliographic record

BibTeX Citation

@article{Batic2026scalarquasinormal,
  author = {Batic, D. and Cornell, A.S. and Dutykh, D.},
  title = {Scalar quasinormal modes of Schwarzschild–anti-de Sitter black holes: spectral analysis and a formal coefficient deformation},
  journal = {Eur. Phys. J. C},
  year = {2026},
  volume = {86},
  number = {10},
  pages = {1252},
  doi = {10.1140/epjc/s10052-026-16449-7},
  abstract = {We study quasinormal modes (QNMs) of a minimally coupled massless scalar field on four-dimensional Schwarzschild–anti-de Sitter (SAdS) black holes using a Chebyshev spectral method. After compactifying the exterior domain, the radial problem is formulated as a quadratic matrix pencil in the dimensionless frequency. For the standard Dirichlet, or vanishing-field, boundary condition at the conformal AdS boundary, we reproduce the known scalar spectra across small, intermediate, and large black holes, including long overtone sequences and the expected approach to pure-AdS normal modes in the small black hole limit. We also impose a one-parameter relation between the slow and fast asymptotic coefficients. For a massless scalar in four dimensions, the slow branch lies outside the standard finite canonical energy domain. Because we neither construct a renormalized variational principle nor prove well-posedness of an evolution containing this branch, we present only the formal spectrum of the deformed radial problem. Each non-Dirichlet value examined yields an additional eigenvalue with positive imaginary part and hence, an exponentially growing separated formal mode. This does not establish a physical instability of the standard scalar theory. The interpolation parameter is representation-dependent. In a black-hole-independent conformal coordinate, the corresponding response coupling scales with the cube of the dimensionless horizon radius, so equal intermediate parameter values for different black-hole sizes do not represent the same boundary coupling.},
  keywords = {quasinormal modes, Schwarzschild–anti-de Sitter black holes, massless scalar field, Chebyshev spectral method, quadratic eigenvalue problem, AdS boundary conditions},
  publisher = {Springer Science and Business Media LLC},
  issn = {1434-6044}
}