Deficiency indices of a one-term symmetric differential operator of even order degenerate in the interior of an interval

Let a(x) element of C∞[-h,h], h>0, be a real function such that a(x)≠0 for x element of [-h,h]. Consider the differential expression sp[f]=(-1)n(xpa(x)f(n))(n) of arbitrary order 2n≥2, which depends on the positive integer p and is degenerate for x=0. Let Hp be the real symmetric operator in L2(-h,h) corresponding to sp[f] and let DefHp be its deficiency index in the upper (or lower) half-plane. The proof of the formula DefHp=2n+p, 1≤p≤n, is presented. It complements the formulae DefHp=2n for p≥2n and DefHp=4n-p for p=2n-2,2n-1 obtained by the same author before.