Quasinormal-Mode Shifts Induced by Internal Structure of Horizonless Compact Objects: A Two-Component Minimal Parameterization and Perturbative Analysis

Description This technical note summarizes a minimal phenomenological analysis of quasinormal‑mode (QNM) frequency shifts induced by small deformations of the external scattering potential around horizonless compact objects (HCOs). The study introduces a two‑component representation of internal structure—“spreading” and “compression”—and combines them into a single effective parameter Kcore. This parameter controls a localized Gaussian deformation of the photon‑sphere potential, motivated by the exponentially decaying anisotropic stresses found in anisotropic TOV models and ECO literature. To evaluate the resulting QNM shifts, the work employs a perturbative formula defined using the Leaver (1985) complex‑mapping normalization, ensuring a mathematically well‑posed inner product for divergent QNM wavefunctions. Numerical calculations for the (ℓ=2,n=0) Schwarzschild mode show a clear linear relation δω∝Kcore, with typical fractional shifts of 10−3–10−4. The note also connects the phenomenological parameter Kcore to concrete quantities in anisotropic TOV models, demonstrating consistency between the minimal parameterization and realistic stellar‑structure profiles. The purpose of this document is to provide a compact, self‑contained reference for the perturbative framework, normalization method, and numerical behavior of QNM shifts caused by small external potential deformations. It is intended as a technical companion to ongoing work on HCO spectroscopy, echo boundary conditions, and internal‑structure modeling.