A note on the glueball mass spectrum

A conjectured duality between supergravity and $N=\infty$ gauge theories gives predictions for the glueball masses as eigenvalues for a supergravity wave equations in a black hole geometry, and describes a physics, most relevant to a high-temeperature expansion of a lattice QCD. We present an analytical solution for eigenvalues and eigenfunctions, with eigenvalues given by zeroes of a certain well-computable function $r(p)$, which signify that the two solutions with desired behaviour at two singular points become linearly dependent. Our computation shows corrections to the WKB formula $m^2= 6n(n+1)$ for eigenvalues corresponding to glueball masses QCD-3, and gives the first states with masses $m^2=$ 11.58766; 34.52698; 68.974962; 114.91044; 172.33171; 241.236607; 321.626549, ... . In $QCD_4$, our computation gives squares of masses 37.169908; 81.354363; 138.473573; 208.859215; 292.583628; 389.671368; 500.132850; 623.97315 ... for $O++$. In both cases, we have a powerful method which allows to compute eigenvalues with an arbitrary precision, if needed so, which may provide a qualitative test for the duality conjecture. Our results matches with the numerical computation of \cite{oog} withing precision reported there in both $QCD_3$ and $QCD_4$ cases. As an additional curiosity, we report that for eigenvalues of about 7000, the power series, although convergent, has coefficients of orders ${10}^{34}$; and therefore tricks are needed to get reliably the function $r(p)$, as also the final answer gets small, of order ${10}^{-6}$ in $QCD_4$. In principle we can go to infinitely high eigenavalues at an expence of computer sufferings, just eventually such computation will slow down to make it inpractical.