Coupled-Channel Spectral Theory for a Non-Separable Geometry: Normal Modes and Two-Boundary Response in the Rotating AdS-Teo Wormhole

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We investigate scalar perturbations of a rotating asymptotically anti-de Sitter (AdS)-Teo traversable wormhole with a controlled non-separable angular deformation. The geometry retains a regular wormhole throat and the required AdS asymptotics, while an explicit quadrupolar deformation generates angular-channel coupling in the intermediate region. A sufficient condition is derived for an ergoregion-free parameter regime in which the spectral analysis is performed. Projecting the geometry-derived Klein–Gordon equation onto spherical harmonics yields a matrix-valued Sturm–Liouville system and an associated quadratic operator pencil. Throat regularity together with normalizable AdS boundary conditions leads to a determinant quantization condition for the discrete normal-mode spectrum. Two-, four-, and six-channel calculations demonstrate systematic numerical convergence of the retained low-lying normal-mode frequencies under enlargement of the angular basis. The complete finite generalized eigenspectrum is also examined without imposing a near-reality selection criterion, and no growing scalar mode is found within the ergoregion-free parameter range and numerical resolutions studied. The six-channel rotation continuation exhibits a finite interior level-repulsion feature accompanied by collective redistribution of the multichannel eigenvectors. The same coupled spectral framework formally defines a matrix-valued two-boundary response whose poles are selected by the normal-mode matching condition; the response plots presented here illustrate this structure using the reduced two-channel effective model rather than a numerical reconstruction of the full six-channel response. We further derive the geometry-dependent short-distance Hadamard subtraction and identify the finite inter-channel structure entering a truncated local quantum mode sum, without claiming a complete numerical evaluation of ⟨Φ2⟩ren. Together, these results establish a geometry-to-spectrum framework in which the spacetime geometry determines the coupled operator, the global boundary conditions determine its normal-mode spectrum, and the same coupled spectral framework organizes the associated asymptotic response and local quantum mode-sum structure.

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