This work revolves around spectral inequalities and their applications in quantum mechanics. We analyze the Schrödinger operator in two dimensions with an attractive potential given by a Bessel–Macdonald function, \(K_0\) . This operator is derived in the non-relativistic approximation of planar quantum electrodynamics \((\textrm{QED}_3)\) models as a framework for evaluation of two-quasi-particle scattering potentials. In particular, this article is concerned with the infimum \(-|E_1|\) of the spectrum of the Schrödinger operator \(H=-\frac{\hbar ^2}{2\mu } \Delta -\alpha V\) in \({\mathbb {R}}^2\) , with V being the spherically symmetric potential function \(K_0\) . Specifically, in the case of angular momentum \(m=0\) , the infimum \(-|E_1|\) is estimated with the help of a weighted version of the Lieb–Thirring inequalities for the moments of its negative eigenvalues. Results are compared with those assuming that the same operator is governed by the usual coulombian potential, 1/r, even in a genuinely two-dimensional space. Then, by using Keller–Lieb–Thirring-type spectral estimates, we obtain an upper and a lower bound for the coupling constant of the Schrödinger operator with the potential given by a Bessel–Macdonald function.
