Analytic treatment of the excited instability spectra of the magnetically charged SU(2) Reissner-Nordstrom black holes

The magnetically charged SU(2) Reissner-Nordstrom black -hole solutions of the coupled nonlinear Einstein -Yang -Mills field equations are known to be characterized by infinite spectra of unstable (imaginary) resonances {omega(n)(r+,r-)}(n=infinity)(n=0:) (here +/- are the black -hole horizon radii). Based on direct numerical computations of the black -hole instability spectra, it has recently been observed that the excited instability eigenvalues of the magnetically charged black holes exhibit a simple universal behavior. In particular, it was shown that the numerically computed instability eigenvalues of the magnetically charged black holes are characterized by the small frequency universal relation omega(n)(r(+) (-)r(-)) =lambda(n,) , are dimensionless constants which are independent of the black -hole parameters. In the present paper we study analytically the instability spectra of the magnetically charged SU(2) Reissner-Nordstrom black holes. In particular, we provide a rigorous analytical proof for the numerically -suggested universal behavior omega(n)(r(+) (-)r(-)) = lambda(n) in the small frequency omega(n)r + << (r(+) (-)r(-)) /(+) regime. Interestingly, it is shown that the excited black hole resonances are characterized by the simple universal relation omega(n)+1/omega(n) = e(-2 pi/root 3). Finally, we confirm our analytical results for the black-hole instability spectra with numerical computations.