We develop a theory of superconducting pairing in low-density Strontium titanate based upon the idea of quadratic coupling of electron density to soft transverse optical phonons [1]. This coupling leads to static attractive potential between electrons which decays
in real space exponentially with the length leff that scales inversely with soft optical gap ωT. For low electron densities n≤1018cm−3 attraction between electrons can be considered local and superconducting transition temperature Tc is calculated using old results [2]. We use independently obtained magnitude g of the quadratic coupling strength and find Tc(n) dependence in agreement with experimental data [3]
for low doping. Next, we show that suppression of Tc by hydrostatic pressure [4] and strong increase of Tc due to isotop substitution 16O→18O observed in [5] are qualitatively explained within our theory.

[1] K. L. Ngai, Phys. Rev. Lett. 32, 215 (1974)
[2] L. P. Gor'kov and T. K. Melik-Barkhudarov, Sov. Phys. JETP 40, 1452-1458 (1961)
[3] X. Lin, G. Bridoux, A. Gourgout, G. Seyfarth, S. Kramer, M. Nardone, B. Fauque, and K. Behnia, Phys. Rev. Lett. 112, 207002 (2014).
[4] C. Enderlein, J. Ferreira de Oliveira, D. A. Tompsett, E. Baggio Saitovitch, S. S. Saxena, G. G. Lonzarich and S. E. Rowley, Nature Comm. (2020), https://doi.org/10.1038/s41467-020-18438-0

[5] A. Stucky, G. W. Scheerer, Z. Ren, D. Jaccard,
J.-M. Poumirol, C. Barreteau, E. Giannini and D. van der Marel, Scientific
Reports, 6:37582 (2016) DOI: 10.1038/srep37582