The asymptotic behavior of the spectrum trace formulas for differential operators with unbounded coefficients
The eigenvalue problem \[ \begin{aligned} u''+vu&=\lambda u,\quad t\in [0,\pi],\tag{1}\\ u(0)&= 0=u(\pi),\tag{2}\end{aligned} \] for \(u\in L^2(0,\pi)\) and \(v\) satisfying \(\int_0^\pi x^\varepsilon (\pi-x)^\varepsilon |v(x)|\, dx<\infty\) for some \(0<\varepsilon<1\), is considered. If \(v=0\) then the eigenvalues are \( \lambda_n=n^2, n\in N,\) with eigenfunctions \(f_n(x)=\sqrt{\frac{2}{\pi}} \sin nx\). Let \[ R_{0,n}(z)h=\sum_{k\neq n} \frac{(h,f_k)}{\lambda_k-z}f_k, \] and \(Vh(x)=v(x)h(x)\). Denoting the eigenvalues of (1)-(2) by \(\mu_n, n\in N\), it is shown that for large \(n\), \[ \mu_n=\lambda_n+\sum_{k=1}^m \alpha_k^{(n)} + O(\gamma_n^mn^{\varepsilon -m(1-\varepsilon)}), \] where \(m\) is the least integer with \(m(1-\varepsilon)>1+\varepsilon\) and \[ \alpha_1^{(n)}= (Vf_n,f_n), \alpha_2^{(n)}=(VR_{0,n}(\lambda_n)Vf_n,f_n), \dots. \] Here \(\delta,\gamma_n>0\) are such that \[ \max_{|z-\lambda_n|\leq n\delta} \| R_{0,n}(z)V\|_{C[0,\pi]}\leq \gamma_n n^{\varepsilon-1} \] and \(\gamma_n\to 0\) as \(n\to \infty\). Similar results are also discussed for more general boundary conditions and for Bessel type equations.
scientific article; zbMATH DE number 5375906