We develop a theory of superconducting pairing in low-density Strontium titanate [1] based upon
the idea of quadratic coupling of electron density to soft transverse optical phonons [2]. 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. For low electron densities n ≤ 10^{18} cm^{−3}
attraction between electrons can be considered local and superconducting transition temperature
Tc is calculated using old results [3]. We use independently obtained magnitude g of the quadratic
coupling strength and find Tc (n) dependence in agreement with experimental data [4] for low doping.
Next, we show that suppression of Tc by hydrostatic pressure [5] and strong increase of T_c due to
isotop substitution ^{16} O →^{18} O observed in [6] are qualitatively explained within our theory.



[1] D.E. Kiselov and M.V. Feigel’man, Phys. Rev. B 104,
L220506 (2021)
[2] K. L. Ngai, Phys. Rev. Lett. 32, 215 (1974)
[3] L.P.Gor’kov and T.K.Melik-Barkhudarov, Sov.Phys. -
JETP 40, 1452-1458 (1961)
[4] Xiao Lin, et al., Phys. Rev. Lett. 112, 207002 (2014).
[5] C. Enderlein, el al., Nature Comm. (2020),
[6] A. Stucky, et al., Scientific Reports, 6:37582 (2016)