Asymptotic behaviour of squares of sum eigenvalues of the singular Sturm-Liouville operator

ABSTRACTIn this thesis , entitled ?Asymtotic Behavior of The Sum of Squares of negativeeigenvalues of The Singular Sturm - Liouville Opereator? , we find asymtotic formulason the sum of squares negative eigenvalues of a self -adjoint operator L generatedby the differantial statement l(y ) = − y′′(x ) − q(x )y(x ) and the boundary candidativey′(0) = 0 in the space L 2 [0, ∞ ) . We assume that the function q(x) , seen in thestatement of l(y), is continous, decresasing and takes positive values in the interval[0, ∞ ) . If λ1 ≤ λ 2 ≤ ... ≤ λ n ≤ ...the negative eigenvalues of the operator L, the weprove that the asymtotic formulas , when ε → +0∑ [1 + O(e )]⋅ ∫ [ ]−1 t0λ2 i = (15π ) q(x ) − ε ⋅ 8q 2 (x ) + 4q (x ) ⋅ ε + 3ε 2 ⋅ dx− λ i < −ε q( x ) ≥ ε∑−1  [ ]∫−βλ2 i = (15π ) 1 + O e − ε q (x ) − ε ⋅ 8q 2 (x ) + 4q ⋅ (x )ε + 3ε 2 ⋅ dx ⋅ − λ i < −ε q( x ) ≥ εhelds under some additional conditions on the function q(x).Keywords : Self - adjoint operator ,Semi bounded operator , Eigenvalue , destict not ofspectrum.