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 $l_{\mathrm{eff}}$ that
scales inversely with soft optical gap $\omega_{T}$. For low electron
densities $n\leq10^{18}cm^{-3}$ attraction between electrons can
be considered local and superconducting transition temperature $T_{c}$
is calculated using old results [2]. We use independently
obtained magnitude $g$ of the quadratic coupling strength and find
$T_{c}(n)$ dependence in agreement with experimental data [3]
for low doping. Next, we show that suppression of $T_{c}$ by hydrostatic
pressure [4] and strong increase of $T_{c}$ due to
isotop substitution $^{16}O\to^{18}O$ 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] Xiao Lin, German Bridoux, Adrien Gourgout, Gabriel
Seyfarth, Steffen Kramer, Marc Nardone, Benoit Fauque, and Kamran
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, "Superconductivity mediated by polar modes in ferroelectric
metals", 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