Summation formula inequalities for eigenvalues of the perturbed harmonic oscillator (Q294163)

The authors study mainly the eigenvalue problem of the form NEWLINENEWLINE\[NEWLINE- u'' (x) + (x^{2} + q (x)) u (x) = \lambda u (x), \quad x \in \mathbb RNEWLINE\]NEWLINE NEWLINEassuming that \(q\) is a non-negative summable potential on the real line. The eigenvalues \(\lambda_{n}\) of the problem satisfy NEWLINE\[NEWLINE\sum_{k = 0}^{n} \biggl( \lambda_{k} - \lambda_{k}^{0} - \frac{1}{\pi \sqrt{ \lambda_{k}^{0}}} \int_{\mathbb R} q (x) d x \biggl) \leq \frac{\chi_{n}}{\pi} \int_{\mathbb R} q (x) d x, \quad n=0,1,\dots,NEWLINE\]NEWLINE where \(\lambda_{k}^{0} = 2 k + 1\) for \(k \in \mathbb N\) are eigenvalues of the quantum harmonic oscillator \(- d^{2} / d x^{2} + x^{2}\), and NEWLINE\[NEWLINE \chi_{n} = \begin{cases} \frac{2 n + 3}{n + 1} \frac{\Gamma (n/2 + 1)}{\Gamma ((n + 1)/2)} - \sum_{k = 0}^{n} \frac{1}{\sqrt{ \lambda_{k}^{0}}}&\text{for }n\text{ odd}, \\ (n + 1) \frac{\Gamma ((n + 1)/2)}{\Gamma (n/2 + 1)} - \sum_{k = 0}^{n} \frac{1}{\sqrt{ \lambda_{k}^{0}}}&\text{for }n\text{ even}.\end{cases}NEWLINE\]NEWLINE Certain generalizations and slightly other type potentials are also considered.