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Consider the operator $L=-\frac{d^2}{dx^2}+q(x)$, where $q(x)$ is the potential with polynomial-type growth, say $|x|^s,s>1$. The eigenvalue problem $$L\phi=\lambda\phi$$ where $\phi$ is in $L^2(\mathbb{R})$ and vanishes at infinity.

We want to study the distribution of the eigenvalues (approximately)?

Classical example: when $s=2$, Hermite orthogonal polynomial comes out.

Is there any analytical result about $s=4$ case?

I am currently looking at books by Titchmarsh , and ODE book by Coddington and Levinson.

Any other classical/modern books papers for reference?

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If $L=-\Delta +|x|^s$, $s>0$, in $R^n$, then $N(\lambda)$, the number of eigenvalues less than $\lambda$, behaves like $\lambda^{N(1/2+1/s)}$ as $\lambda \to \infty$. This is in Titchmarsh "Eigenfunction expansions...Part II, Sect. 17.8 or in Reed Simon Section XIII.

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  • $\begingroup$ Any theorems about the locations of those eigenvalues? Without knowing any (fundamental) solutions $\endgroup$
    – DuFong
    Mar 5, 2020 at 2:51
  • $\begingroup$ I found those locations result in Titchmarsh VII, 7.1.7, thanks for the hint. $\endgroup$
    – DuFong
    Mar 5, 2020 at 3:25
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It depends on what you mean by "analytical result". Even quartic oscillator $q(x)=x^4+ax^2$ was studied VERY much. See, for example,

Bender, Carl M.; Wu, Tai Tsun Anharmonic oscillator. Phys. Rev. (2) 184 (1969), 1231–1260.

Simon, Barry Coupling constant analyticity for the anharmonic oscillator. (With appendix). Ann. Physics 58 (1970), 76–136.

Voros, A. The return of the quartic oscillator: the complex WKB method. Ann. Inst. H. Poincaré Sect. A (N.S.) 39 (1983), no. 3, 211–338.

Eremenko, Alexandre; Gabrielov, Andrei Analytic continuation of eigenvalues of a quartic oscillator. Comm. Math. Phys. 287 (2009), no. 2, 431–457,

Giachetti, Riccardo; Grecchi, Vincenzo Bender-Wu singularities. J. Math. Phys. 57 (2016), no. 12, 122109.

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