Sharp conditions on the quantization data and sharp contraction rates for the Voros–Avila exact quantization scheme

For the one-dimensional homogeneous anharmonic oscillator (Schrödinger operator with potential $q^{2M}$), Voros showed that the parity-split spectrum is a fixed point of an explicit nonlinear transformation $T_{\theta,Q}$ of sequences, and Avila (Comm. Math. Phys. 249 (2004) 305–318) proved that, under the hypotheses $Q_k=k+O(1)$ and $Q_k>(k-\tfrac{1}{2})\theta/\pi$, this fixed point is globally attractive in the properly normalized growth class and exponentially attractive in polynomial-weight norms — settling the convergence of the exact quantization scheme. The quantitative theory, however, is incomplete: the weakest hypothesis of the second type that is necessary for existence, uniqueness and attractivity is unknown; the spectral radius of the linearization $DT(P)$ in the weighted spaces is only bounded below (by $S_\epsilon/S_0$) and not computed; and all known contraction constants degenerate in the limit $\theta\to\pi$ (oscillator degree $M\to\infty$, $\alpha_\theta\to 2$), where the scheme's efficiency is unquantified. The problem asks for the sharp quantitative completion of this theory. The subsequent literature treats Avila's theorem as settled and does not address these questions.

Background

The one-dimensional homogeneous anharmonic oscillator is the Schrödinger operator $H=-d^2/dq^2+q^{2M}$ on $L^2(\mathbb{R})$ with $M=2,3,\ldots$; it has a discrete spectrum $0<E_0<E_1<\cdots$ with polynomial growth $E_k\sim\nu k^{2M/(M+1)}$. Ordinary semiclassical (WKB) theory assigns energy levels through the Bohr–Sommerfeld quantization condition — the rule that a bound state corresponds to a half-integer number of oscillations of the classical momentum over the classically allowed region — which for anharmonic potentials yields only an asymptotic expansion in powers of $1/k$ whose series is factorially divergent, so it cannot be turned into a convergent computational scheme.

Voros's exact quantization scheme (developed through his exact-WKB completion of the WKB expansion, which resums its factorially divergent series) replaces the asymptotic rule by an exact fixed-point problem. Split the spectrum by parity, $P^{\rm even}_i=E_{2i-2}$, $P^{\rm odd}_i=E_{2i-1}$. Fix $0<\theta<\pi$ and define, for positive numbers $E,E'$, the phase $\theta(E',E)=\arctan(\sin\theta/(E'E^{-1}+\cos\theta))$, taken in the branch with values in $(0,\pi)$ (it is the argument of the complex number $E'E^{-1}+e^{i\theta}$), and for sequences $X,Y$ of positive reals set $\phi_j(X,Y)=\frac{1}{\pi}\sum_{k\ge 1}\theta(X_k,Y_j)$. The exact quantization operator $T\equiv T_{\theta,Q}$ is defined implicitly by $\phi(X,T(X))=Q$. Voros proved that $P^{\rm even}$ and $P^{\rm odd}$ are fixed points of $T_{\theta,Q^{\rm even}}$ and $T_{\theta,Q^{\rm odd}}$ respectively, where $\theta=\frac{M-1}{M+1}\pi$ and $Q^{\rm even}_k=k-\frac{3}{4}+\frac{M-1}{4(M+1)}$, $Q^{\rm odd}_k=k-\frac{1}{4}-\frac{M-1}{4(M+1)}$. Because $T$ commutes with positive dilatations ($T(\lambda X)=\lambda T(X)$), the fixed-point equation alone does not pin down the spectrum; a growth normalization $P_k\sim 2^{\alpha_\theta}\nu k^{\alpha_\theta}$ with $\alpha_\theta=1+\theta/\pi=\frac{2M}{M+1}$ is needed to select the physical solution. In Avila's paper both spectral inputs — the growth asymptotics (its Proposition 1.1) and the fixed-point identities (its Proposition 1.2) — are cited to Voros's earlier work, and the paper's own analysis then proceeds entirely in sequence space.

Avila proved the convergence of the resulting iteration scheme in the natural sequence spaces: writing $U^0(\alpha_\theta,0)$ for the class $X_k=e^{o(1)}k^{\alpha_\theta}$ and $U(P,\epsilon)$ for $X_k=P_k+O(k^{\alpha_\theta-\epsilon})$ (in logarithmic coordinates, $u(\epsilon)$ with norm $\|v\|_\epsilon=\sup_k k^\epsilon|v_k|$), he showed under the hypotheses (2.1) $Q_k=k+O(1)$ and (2.2) $Q_k>(k-\tfrac{1}{2})\theta/\pi$ that $T_{\theta,Q}$ has a unique fixed point $P$ in $U^0(\alpha_\theta,0)$; that $P$ is a global attractor there; and that $P$ is exponentially attractive in each weighted space $U(P,\epsilon)$, $0<\epsilon<2$. The analysis passes through the drift functional $D_\alpha=\sin(\theta/\alpha)/\sin(\pi/\alpha)$ (the exponential scale-drift per iteration of sequences in the growth class of exponent $\alpha$; the unique $\alpha>1$ with $D_\alpha=1$ is the critical exponent $\alpha_\theta$, the only growth rate at which a fixed point can live) and through the linearization $DT$, an infinite positive stochastic matrix (rows summing to $1$) whose contraction in the weighted norms is governed by

$$S_\epsilon=\int_0^\infty\frac{s^{-\epsilon}}{s^{\alpha_\theta}+2\cos\theta+s^{-\alpha_\theta}}\,ds=\frac{\pi}{\alpha_\theta\sin\theta}\frac{\sin((1-\epsilon)\theta/\alpha_\theta)}{\sin((1-\epsilon)\pi/\alpha_\theta)},$$

with minimum contraction factor $S_1/S_0=\alpha_\theta-1$ at $\epsilon=1$. Two final remarks of the paper settle the negative boundary: the fixed point is not exponentially attractive in the unweighted $U^0(\alpha_\theta,0)$ metric (explicit counterexamples converge arbitrarily slowly), and the line of fixed points $\lambda P$ is not a global attractor in $U(\alpha_\theta,0)$. The paper also notes that hypothesis (2.1) can be relaxed to $Q_k=e^{o(1)}k$, that if $\sum_{j\le k}Q_j\le\theta k^2/(2\pi)$ for some $k$ then no fixed point exists at all, and that as $\theta\to\pi$ (i.e. $M\to\infty$, $\alpha_\theta\to 2$) the contraction factor becomes weak.

What is not settled, and is the subject of this problem, is the sharp form of these statements: the weakest condition of type (2.2) on the quantization data $Q$ that the convergence theorem truly needs — the paper itself observes (§2.1, after (2.3)) that, since $\sum_{j\le k}Q_j\le\theta k^2/(2\pi)$ for some $k$ precludes any fixed point, "some condition (possibly weaker) in the line of our second condition is necessary for our results to hold"; the exact spectral radius (the modulus of the largest spectral value) of $DT(P)$ on each weighted space, for which only the lower bound $S_\epsilon/S_0$ is known (Remark 2.3); the borderline behavior as $\epsilon\to 0$ or $\epsilon\to 2$ where the known contraction constants degenerate to $1$; and effective estimates in the degenerate regime $\theta\to\pi$, where Remark 2.3 records that the contraction factor becomes weak.

Problem Statement

Let $0<\theta<\pi$, let $Q=(Q_k)_{k\ge 1}$ be a sequence of real numbers, and let $T_{\theta,Q}$ be the operator on sequences of positive reals defined implicitly by $\phi(X,T(X))=Q$, where $\theta(E',E)=\arctan(\sin\theta/(E'E^{-1}+\cos\theta))$ is taken in the branch with values in $(0,\pi)$ (the argument of the complex number $E'E^{-1}+e^{i\theta}$) and $\phi_j(X,Y)=\frac{1}{\pi}\sum_{k\ge 1}\theta(X_k,Y_j)$. Write $U^0(\alpha_\theta,0)$ for the growth class $X_k=e^{o(1)}k^{\alpha_\theta}$ with $\alpha_\theta=1+\theta/\pi$, and $U(P,\epsilon)$ for $X_k=P_k+O(k^{\alpha_\theta-\epsilon})$; in logarithmic coordinates $x_k=\ln X_k$ the infinitesimal counterpart is the weighted space $u(\epsilon)$, the space of sequences $v_k=O(k^{-\epsilon})$ with norm $\|v\|_\epsilon=\sup_k k^\epsilon|v_k|$, on which the linearization $DT(P)$ acts. Avila's theorem states: if $Q_k=k+O(1)$ (hypothesis (2.1), relaxable to $Q_k=e^{o(1)}k$) and $Q_k>(k-\tfrac{1}{2})\theta/\pi$ (hypothesis (2.2)), then $T_{\theta,Q}$ has a unique fixed point $P$ in $U^0(\alpha_\theta,0)$, which is a global attractor there and is exponentially attractive in each $U(P,\epsilon)$, $0<\epsilon<2$, in the sense that $\sup_k k^{-\alpha_\theta+\epsilon}|T^n(X)_k-P_k|\le C\lambda^n$ for some $\lambda=\lambda(\epsilon)<1$; and if $\sum_{j\le k}Q_j\le\theta k^2/(2\pi)$ for some $k$ then no fixed point exists. The sharp quantitative completion of this theory concerns the following open questions. The first is the direction formulated in the paper itself (§2.1, after (2.3)); the second and third sharpen the two caveats recorded in its Remark 2.3, namely that only a lower bound is available for the spectral radius of $DT(P)$ and that the contraction factor becomes weak as $\theta\to\pi$:

(1) Sharp hypotheses: determine the weakest condition of the type of (2.2) on $Q$ — ideally a necessary and sufficient condition — under which the conclusions of Avila's theorem hold (existence of a fixed point in $U^0(\alpha_\theta,0)$ under the relaxed (2.1), uniqueness, and global attractivity); in particular decide whether the constant $\tfrac{1}{2}$ and the affine form $k\theta/\pi$ in (2.2) can be improved.

(2) Sharp rates: for quantization data $Q$ satisfying (2.1) and (2.2), with $P=P_{\theta,Q}$ the corresponding fixed point, determine the exact spectral radius of the linearization $DT(P)$ as an operator on each weighted space $u(\epsilon)$ (at least for $0<\epsilon<2$), deciding whether the known lower bound $S_\epsilon/S_0$ (with $S_\epsilon=\int_0^\infty\frac{s^{-\epsilon}}{s^{\alpha_\theta}+2\cos\theta+s^{-\alpha_\theta}}\,ds$, so that $S_1/S_0=\alpha_\theta-1$) is attained. That lower bound is independent of $Q$, and the paper's contraction estimates are proved uniformly for $X$ ranging over uniformly bounded subsets of $U^0(\alpha_\theta,0)$ (its Lemma 2.13); the question is correspondingly whether the spectral radius equals $S_\epsilon/S_0$ uniformly for admissible $Q$, or whether a different value occurs. Describe also the borderline behavior of the optimal contraction constants as $\epsilon\to 0$ and $\epsilon\to 2$, where exponential attraction degenerates into the known arbitrarily slow convergence at $\epsilon=0$.

(3) Degenerate regime: obtain effective (uniform or asymptotically sharp) contraction estimates as $\theta\to\pi$, equivalently oscillator degree $M\to\infty$ with $\alpha_\theta\to 2$ and minimum factor $\alpha_\theta-1\to 1$, quantifying the efficiency of the exact quantization scheme for very large $M$; as in the paper's own estimates, the uniformity requested is with respect to $\theta$ near $\pi$ for admissible $Q$.

Each of (1)–(3) can be resolved independently, and the sharp quantitative completion of the theory is their conjunction.

References

1. Artur Avila, "Convergence of an Exact Quantization Scheme", Communications in Mathematical Physics 249 (2004), no. 2, 305–318. DOI: 10.1007/s00220-004-1112-9. https://doi.org/10.1007/s00220-004-1112-9 (author e-print arXiv:math/0306218)
2. André Voros, "Exact Anharmonic Quantization Condition (In One Dimension)", in: Quasiclassical Methods (Minneapolis, MN, 1995), The IMA Volumes in Mathematics and its Applications 95, Springer, New York, 1997, pp. 189–224. DOI: 10.1007/978-1-4612-1940-8_10. https://doi.org/10.1007/978-1-4612-1940-8_10
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