Trace Formula for Sturm--Liouville Operators with Singular Potentialsстатья
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Аннотация:Suppose that u(x) is a function of bounded variation on the closed interval [0,π], continuous at the endpoints of this interval. Then the Sturm—Liouville operator Sy=−y″+q(x) with Dirichlet boundary conditions and potential q(x)=u′(x) is well defined. (The above relation is understood in the sense of distributions.) In the paper, we prove the trace formula
∑k=1∞(λ2k−k2+b2k)=−18∑h2j,bk=1π∫π0coskxdu(x), where the λk are the eigenvalues of S and h j are the jumps of the function u(x). Moreover, in the case of local continuity of q(x) at the points 0 and π the series ∑∞k=1(λk−k2) is summed by the mean-value method, and its sum is equal to
−(q(0)+q(π))4−18∑h2j.