Conside the One dimentional Schrodinger Operator

$$ -\frac{d^2}{dx^2} + ( V(x) + E ) $$

Where the Potential Function $V$ is of the form $V(x) = ax^2 + b^2x^4$ , $a,b \in \mathbb{R} $. What is known about the Spectral growth of the above Operator for large enough value of $E$,$|a|$ and $b$.Can we describe the singular values of the Operator as a function of $E$,$a$ and $b$ upto some bounded error term?
Thanks in advance for any comment,suggestion or reference in that direction.


1 Answer 1


First of all, it is sufficient to consider only one parameter: making a change of the independent variable $z\mapsto kz$, with appropriate $k$ one can eliminate either $a$ or $b$. Let us eliminate $b$ and consider $$-y''+(x^4+ax^2)y=\lambda y.$$ Then eigenvalues (in $L^2(R)$) become functions of $a$, and the asymptotics is $$\lambda_n\sim cn^{4/3},$$ where $c$ is an absolute constant (it does not depend on $a$). One can write several terms of asymptotic expansion in decreasing powers of $n$, coefficients of these further terms will depend on $a$. In fact one can write and prove an infinite asymptotic expansion, but the resulting series is divergent.

(In the general case of a polynomial potential, $V(x)=x^m+\ldots,\;x\to\infty$, the order of growth of eigenvalues is $2m/(m+2)$, see Sibuya and Shin below).

These functions $\lambda_n(a)$ were very much studied, I mention first of all the paper:

C. Bender and T. Wu, Anharmonic oscillator. Phys. Rev. (2) 184 1969 1231–1260.

This paper has more than 1000 references on Google scholar. By looking in these references you can obtain a more or less complete picture of what is known. (The situation is roughly like this: all statements in this paper are correct, and it contains a very complete discussion of these eigenvalues. However most of the things are proved on the "physical level of rigor", or just illustrated by computation. In the subsequent papers most of these statements were rigorously justified.) If you are only asking about asymptotic expansion for real $a$, it is written explicitly in Bender-Wu, and it is easy to justify. The difficult thing is proving the global analytic properties of $\lambda_n(a)$, for complex $a$. These functions are very complicated, indeed. Some principal mathematical papers which discuss them are:

B. Simon, Coupling constant analyticity for the anharmonic oscillator, Ann. Phys., 58 (1970) 76–136.

A. Voros, The return of the quartic oscillator, Annales de l'Institut Henri Poincare. Section A, Physique Theorique; v. 39(3); 1983, p. 211-338.

B. Simon, Large order and summability of eigenvalue perturbation theory: a mathematical overview, Intl. J. Quantum Chemistry, 21 (1982) 3–25.

E. Delabaere, F. Pham, Unfolding the quartic oscillator, Ann. Physics 261 (1997), no. 2, 180–218.

J. Loeffel and A. Martin, Proprietes analytiques des niveaux de l’oscillateur anharmonique et convergence des approximants de Pade, dans le livre: Cargese Lectures in Physics, vol. 5, Gordon and Breach NY, 1972, 415–429.

Kwang C. Shin, Schrodinger type eigenvalue problems with polynomial potentials: asymptotics of eigenvalues, arXiv:math.SP/0411143v1.

A. Eremenko and A. Gabrielov, Analytic continuation of eigenvalues of a quartic oscillator, Comm. Math. Phys., v. 287, No. 2 (2009) 431-457.

Y. Sibuya, Global theory of a second order linear differential equation with a polynomial coefficient, North Holland, 1975.

EDIT. The leading term of the asymptotics $n^{4/3}$ is not difficult to obtain. By WKB there exists a unique solution satisfying $$y(x,\lambda)=(1+o(1))x^{-1}\exp\left(-\frac{1}{3}x^3\right),\; x\to+\infty.$$ For every $x$ this is an entire function of $\lambda$, and $$y(0,\lambda)= \exp\left(K\lambda^{3/4}-(1/4)\log\lambda+o(1)\right).$$ Zeros of this function or zeros of the derivative $(d/dx)y(x,\lambda)$ at $x=0$ are eigenvalues. From this the asymptotics of eigenvalues are easily obtained.

  • $\begingroup$ In the Potential function $V$ ,$a$(the coefficient of $x^2$) can be highly negative as well,so in that case the differential equation you took would be of the form $-y''+(x^4+ax^2)y=\lambda y$,So,I really wanted to know the singular value distribution of this differential equation for $|a|$ large enough.Could you please help me on that?Also,in the paper 'C. Bender and T. Wu, Anharmonic oscillator. Phys. Rev. (2) 184 1969 1231–1260',I think they gave some asymptotic expansion of the Ground state energy.So,could you elaborate more about the growth $\lambda_n\sim cn^{4/3}$? $\endgroup$
    – Surajit
    Mar 20, 2017 at 6:47
  • $\begingroup$ Whether $a$ is positive or negative, does not matter for asymptotics of (very large) eigenvalues. For the rate of growth look in the paper of Shin, for example. $\endgroup$ Mar 20, 2017 at 14:05
  • $\begingroup$ Thanks you very much for the details.Now it is clear to me. $\endgroup$
    – Surajit
    Mar 20, 2017 at 18:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy