We present a method for comparing the classical and quantum calculations of the electric quadrupolar Love number $k_2$ and show that our previous derivation of the quantum Love number of a quantum blackhole matches exactly the classical calculation of $k_2$ when quantum expectation values are replaced by the corresponding classical quantities, as dictated by the Bohr correspondence principle. The standard derivation of $k_2$ for classical relativistic stars relies on fixing boundary conditions on the surface of the star for the Einstein equations in the presence of an external perturbing field. An alternative method for calculating $k_2$ uses properties of the spectrum of the non-relativistic fluid modes of the star. We adopt this alternative method and use it to derive an effective description of the interior modes in terms of a collection of driven harmonic oscillators characterized by different frequencies and amplitudes. We compare these two classical methods and find that most of the interior information can be integrated out, reducing the problem of calculating $k_2$ to fixing a single boundary condition for the perturbed Einstein equations on the surface of the deformed star. We then determine this single boundary condition in terms of the spectrum of the object and proceed to identify the relationship between classical quantities and quantum expectation values and to verify the agreement between the results of the effective classical calculation and the quantum calculation.
Institut des Hautes Études Scientifiques（高等科学研究所）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（理论科学国际中心，塔塔基础研究所）
Love at First Loop: Quantum Deformability of Near-Extremal Black Holes
Universidad de Murcia（穆尔西亚大学）；CERN（欧洲核子研究组织）；KU Leuven（荷语鲁汶大学）；Leuven Gravity Institute, KU Leuven（荷语鲁汶大学引力研究所）；Institució Catalana de Recerca i Estudis Avançats (ICREA)（加泰罗尼亚高级研究学院）；Universitat de Barcelona（巴塞罗那大学）
Instituto de Ciencias Exactas y Naturales (ICEN), Universidad Arturo Prat（阿图罗·普拉特大学精确科学与自然学院）；Facultad de Ciencias, Universidad Arturo Prat（阿图罗·普拉特大学理学院）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究所国际理论科学中心）
Extreme Love in the SPA: constraining the tidal deformability of supermassive objects with extreme mass ratio inspirals and semi-analytical, frequency-domain waveforms
Dynamical Tidal Response of Schwarzschild Black Holes
Université Paris Cité（巴黎西岱大学）；The University of Chicago（芝加哥大学）；Utah State University（犹他州立大学）；Harvard University（哈佛大学）；Universidad Autonoma de Madrid（马德里自治大学）
