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An evaluation of powers of the negative spectrum of Schrödinger operator equation with a singularity at zero. (English) Zbl 1378.34082
Summary: In this study, we investigate the discreteness and finiteness of the negative spectrum of the differential operator \(L\) in the Hilbert space \(L_{2} (H, [0,\infty)) \), defined by \[ Ly=- \frac{d^{2} y}{d x^{2}} + \frac{A(A+I)}{x^{2}} y-Q(x)y \] under the boundary condition \(y (0) =0\).
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.
34G10 Linear differential equations in abstract spaces
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
Full Text: DOI
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