An evaluation of powers of the negative spectrum of Schrödinger operator equation with a singularity at zero

Abstract In this study, we investigate the discreteness and finiteness of the negative spectrum of the differential operator L in the Hilbert space L 2 ( H , [ 0 , ∞ ) ) $L_{2} ( H,\ [ 0,\ \infty))$ , defined as L y = − d 2 y d x 2 + A ( A + I ) x 2 y − Q ( x ) y $Ly=- \frac{d^{2} y}{d x^{2}} + \frac{A(A+I)}{x^{2}} y-Q(x)y$ , under the boundary condition y ( 0 ) = 0 $y ( 0 ) =0$ . In the case when the negative spectrum is finite, we obtain an evaluation for the sums of powers of the absolute values of negative eigenvalues. The obtained result is applied to a class of equations of mathematical physics