The second regularized trace formula for the Sturm-Liouville problem with spectral parameter in a boundary condition (Q2722108)

The authors consider the equation NEWLINE\[NEWLINE -y''+q(x)y=\lambda y,\quad 0\leq x\leq 1, \tag{1} NEWLINE\]NEWLINE where \(q\) is a real-valued function from \(C^3[0,1]\) such that NEWLINE\[NEWLINE q(1)=q'(1)=q'(0)=\int_0^1q(x) dx=0, NEWLINE\]NEWLINE with the boundary conditions NEWLINE\[NEWLINE y'(0)=0,\quad -y(1)=\lambda y'(1).\tag{2}NEWLINE\]NEWLINE A regularization of the second trace \(\sum (\lambda_n^2-\mu_n^2),\) where \(\{ \lambda_n\}\) is the spectrum of problem (1)--(2) with the given \(q\), and \(\{ \mu_n\}\) is the spectrum of (1)--(2) with \(q=0\), is defined and computed explicitly.