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Summation formula inequalities for eigenvalues of the perturbed harmonic oscillator. (English) Zbl 1347.34126

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

MSC:

34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
34B09 Boundary eigenvalue problems for ordinary differential equations
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References:

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