The deficiency index of a symmetric ordinary differential operator with complex coefficients (Q1235816)

Theorem. Suppose \(0<r_2<r_1\). Suppose \[ P_0 = x^3, \] \[ P_1 = x^3(-3r_1 - r_2), \] \[ P_2 = x^3 [(3r_1^2 + 3r_1r_2) + 27x^{-2}] , \] \[ P_3 = x^3 [-(r_1^3 + 3r_1^2r_2 ) - 18(3r_1 + r_2)x^{-2}], \] \[ P_4 = x^3 [r_1^3r_2 + 12(3r_1^2 + 3r_1r_2)x{-2}], \] \[ P_5 = x^2 [-6(r_1^3 + 3r_1^2r_2 )x{-1}], \] \[ P_6 = x [3r_1^2r_2], \] \[ P_7 \equiv 0.\] Let \(Ly = \sum_{k=0}^7 i^k L_ky\), where \[ L_{2r}y= \{P_{7-2r}y^{(r)}\}^{(r)},\] \[L_{2r+1}y= \tfrac12 \{P_{n-2r-1}y^{(r)}\}^{(r+1)} + P_{n-2r-1}y(r+1) )^{(r)}\}. \] Then \(L\) has deficiency index \((3,5)\). The proof uses the theory of regular and irregular singular points for a linear system \(y' = A(x)y\), \(A(x)\) analytic in a sector, to obtain asymptotic formulas for a fundamental set of solutions for \(Ly = \lambda y\).