An approximate regularized trace formula for an ordinary fourth-order differential operator (Q1873475)

Let \(T\) be a positive self-adjoint operator in a separable Hilbert space such that \(T^{-1/2}\) is a trace class operator and let \(\{\lambda_n\}\), \(n\geq 1\), be the eigenvalues of \(T\). Let \(\{\mu_n\},\) \(n\geq 1\), be the eigenvalues of the perturbation \(T+V\), where \(V\) is a bounded self-adjoint operator. Suppose also that \(T^{-1/2}=A+B,\) where \(A\) is a trace class Volterra operator and \(B\) is finite-dimensional. The authors prove that \[ \sum_{n=1}^\infty (\mu_n^{1/2}-\lambda_n^{1/2})=\frac{1}{2} Sp(AV)+ \frac{1}{2} Sp(BV)+ r(T,V), \] \[ | r(T,V)| \leq 2^{-1}\| V\| ^2 \| T^{-1/2}\| \| T^{-1/2}\| _1. \] This result is applied to the case of a boundary value problem for the fourth-order ordinary differential operator \(\frac{d^4}{dx^4} +q(x).\)