## 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:

  M. Abramowitz and I.A. Stegun: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series 55 , For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC, 1964.  E.V. Aleksandrova, O.V. Bochkareva and V.E. Podol’skiĭ: Summation of regularized traces of the singular Sturm-Liouville operator , Differential Equations 33 (1997), 287-291. · Zbl 0912.34026  C. Bandle: Isoperimetric Inequalities and Their Applications, Monographs and Studies in Mathematics 7 , Pitman, Boston, Mass., 1980. · Zbl 0436.35063  L.A. Dikiĭ: On a formula of Gel’fand-Levitan , Uspehi Matem. Nauk (N.S.) 8 (1953), 119-123.  P. Freitas and J.B. Kennedy: Summation formula inequalities for eigenvalues of Schrödinger operators , · Zbl 1419.11058  B.M. Levitan and I.S. Sargsjan: Sturm-Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series) 59 , Kluwer Acad. Publ., Dordrecht, 1991.  A. Pushnitski and I. Sorrell: High energy asymptotics and trace formulas for the perturbed harmonic oscillator , Ann. Henri Poincaré 7 (2006), 381-396. · Zbl 1087.81019  V.A. Sadovnichiĭ and V.E. Podol’skiĭ: Traces of operators , Uspekhi Mat. Nauk 61 (2006), 89-156, (Russian), translation in Russian Math. Surveys 61 (2006), 885-953.  G. Szegő: On an inequality of P. Turán concerning Legendre polynomials , Bull. Amer. Math. Soc. 54 (1948), 401-405. · Zbl 0032.27502  G. Szegő: Orthogonal Polynomials, fourth edition, Amer. Math. Soc., Providence, RI, 1975.  E.C. Titchmarsh: The Theory of the Riemann Zeta-Function, second edition, Oxford Univ. Press, New York, 1986.
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