Consistency of Dalgarno–Lewis perturbation theory for confining two-bodys ystems: second-order corrections in the Cornell potential

Confining two-body systems provide a useful setting for examining the internal consistency ofperturbative treatments of bound-state wavefunctions. In this work, we investigate theapplicability of Dalgarno--Lewis perturbation theory beyond first order within aCornell-type potential,\( V(r)=br-\tfrac{4\alpha_s}{3r}+c \),treating the linear confinement term as the parent Hamiltonian and incorporating the Coulombicinteraction perturbatively so as to retain analytic control over the solutions. Going beyond standard first-order implementations, we carry out a systematic second-orderDalgarno--Lewis correction to the Airy-function ground state and analyze its impact onwavefunction moments. We show that observables involving higher spatial moments are notstabilized at first order, indicating a breakdown of first-order Dalgarno--Lewis theory for suchquantities in confining two-body systems. Inclusion of second-order contributions is thereforerequired for internal consistency. As an explicit application, we evaluate effective overlap functions, their slopes and curvatures,as well as geometric and electromagnetic radii for representative heavy--light mesons($D$, $D_s$, $B$, and $B_s$). While the analysis is model dependent and does not rely on strictheavy-quark universality, the resulting magnitudes and systematic trends are compatible withthose obtained in relativistic quark models, QCD sum-rule approaches, and continuumDyson--Schwinger/Bethe--Salpeter studies. The present results provide semi-analytic benchmarksthat clarify the role of higher-order wavefunction corrections in few-body bound-state problems. PACS numbers: 12.39.Pn, 12.39.Jh, 14.40.Lb, 14.40.Nd