Rigorous and effective completion of Parisi's trace identities for the one-dimensional Schrödinger operator with power-law potentials

For the one-dimensional Schrödinger operator $H=p^{2}/2+x^{q}$ with $q$ a positive even integer (and $p=-i\hbar\,d/dx$, $\hbar$ the semiclassical parameter), Parisi (1982) derived trace identities linking the spectral zeta function $Z(s)=\mathrm{Tr}\,H^{-s}$, the short-time heat-trace expansion coefficients $B_k$, and the coefficients $C_i$ of the WKB eigenvalue asymptotics, and used them numerically to estimate low-lying eigenvalues of the quartic oscillator. The derivation was formal, the numerics were tested only at two WKB orders for $q=4$, and no general formula for the $B_k$ was given. Later literature (exact WKB, ODE/IM correspondence, with Avila's convergence proof in the homogeneous case) has rigorously established equivalent and stronger sum rules for homogeneous potentials, so the surviving open core is: a fully rigorous treatment with effective quantitative control — remainder bounds and domains of validity uniform in the semiclassical parameter and in the continuation variable $s$, a convergence and error analysis of the truncation scheme that combines retained exact eigenvalues with WKB approximants for arbitrary even $q$, and rigorous arbitrary-order control of the coefficients $B_k$ and of their dependence on the WKB coefficients $C_i$.

Background

The anharmonic oscillator $H=p^{2}/2+x^{q}$ (with $q$ a positive even integer, so the potential confines and the spectrum $\{\lambda_n\}_{n\ge 0}$ is discrete and positive) is a basic laboratory of semiclassical physics. The WKB method (Wentzel–Kramers–Brillouin, the semiclassical approximation constructing wavefunctions from the classical momentum) gives the large-$n$ eigenvalue asymptotics $\lambda_n\sim\hbar^{2q/(q+2)}N^{2q/(q+2)}(C_0+C_1N^{-2}+C_2N^{-4}+\cdots)$ with $N=n+\tfrac12$; the coefficients $C_i$ are computable by an explicit recurrence (Bender–Olaussen–Wang 1977), but the series is divergent and its accuracy for the lowest levels is poor. Two exact spectral objects encode the full spectrum: the heat trace $G(t)=\mathrm{Tr}\,e^{-tH}$, whose short-time expansion $G(t)\sim\sum_k B_k\hbar^{2k-1}t^{-s_k}/\Gamma(1-s_k)$ defines the heat-trace coefficients $B_k$, and the spectral zeta function $Z(s)=\sum_n\lambda_n^{-s}=\mathrm{Tr}\,H^{-s}$, initially convergent only for large $\mathrm{Re}\,s$ and extended meromorphically in $s$. Classical trace identities for one-dimensional Schrödinger operators go back to Gelfand and Dikii in the context of the Korteweg–de Vries (KdV) equation, where traces of resolvent powers are expressed through integrals of the potential. Parisi's 1982 paper (a 1978 preprint of the Laboratoire de Physique Théorique de l'École Normale Supérieure, LPTENS 78/9, published in Lecture Notes in Mathematics 925 and reprinted in his 1992 World Scientific volume) extended this circle of ideas to confining monomial potentials: using the Mellin transform $M(s)$ of the resolvent trace he obtained $Z(s)=\tfrac{1}{\pi}\sin(\pi s)M(s)$, deduced from the WKB growth law that $Z(s)$ is meromorphic with simple poles only at $s_k=(\tfrac12+\tfrac1q)(1-2k)$, showed that the residues satisfy $Z_k=\tfrac{\sin(\pi s_k)}{\pi}B_k$ with $B_k$ algebraic functions of the WKB coefficients $C_i$ (the derivation used a formal short-time heat-kernel expansion organized by Baker–Campbell–Hausdorff ordering of operator exponentials, i.e. the formula reordering exponentials of noncommuting operators), and finally subtracted WKB approximants $W_n^{(j)}=\hbar^{2q/(q+2)}N^{2q/(q+2)}\sum_{i\le j}C_iN^{-2i}$ from the zeta sum to obtain convergent trace identities that determine the lowest eigenvalues; for $q=4$ this gave the quartic ground state to about $10^{-4}$ (literature values from Hioe–Montroll 1975 and Bender–Olaussen–Wang 1977 served as reference). What the paper did not supply — the further-work program this problem records — is: rigorous control of the formal expansions and of the analytic continuation in $s$ (remainders, domains of validity, uniformity in $\hbar$ versus continuation in $s$); a quantitative stability, convergence-rate, and error analysis of the truncation scheme for arbitrary even $q$ and higher accuracy targets; and an explicit general formula or efficient algorithm generating $B_k$ to high order for arbitrary even $q$. The modern framework that partially settled these questions is the exact WKB method (which treats the divergent WKB series by Borel resummation — summing the divergent series through its Borel transform — to obtain exact results; it was developed for polynomial potentials by Voros and extended by Kawai and Takei to broader classes of differential equations, with Écalle's resurgence theory, the framework in which divergent expansions and the singularities of their Borel transforms are organized into a mutually consistent exact structure), the resulting exact functional relations for spectral determinants $D_N(\lambda)=\exp[-\partial_s Z(s,\lambda)|_{s=0}]$, and the ODE/IM correspondence (which identifies spectral functions of certain ordinary differential equations with quantities of integrable quantum field theory); Avila (2004) proved convergence (contractivity) of the exact quantization scheme in the homogeneous case, and Steiner (1985) derived closed-form 'magic sum rules' for confinement potentials $V(r)=g\,r^{p}$ with coupling $g>0$ and arbitrary real exponent $p>0$ (which includes non-even exponents). Within this framework the gaps are explicit: full resurgence for the quartic-oscillator WKB solutions and regularity and contractivity of the exact-quantization map beyond the homogeneous-potential case settled by Avila remain unproved, and current numerical practice still evaluates spectral zeta values by combining a small number of accurately computed low-lying levels with WKB eigenvalues for the tail without a quantitative error analysis.

Problem Statement

Put Parisi's trace identities for the one-dimensional Schrödinger operator $H=p^{2}/2+x^{q}$ ($q$ a positive even integer; results are required for all positive even $q$, with an optional extension to power-law confinement $|x|^{q}$ for general real exponent $q>0$) on a fully rigorous and effective footing, completing the program the 1982 paper left formal. Concretely: (1) Rigorous justification — give a rigorous proof of the short-time expansion of the heat trace $G(t)=\mathrm{Tr}\,e^{-tH}$, namely $G(t)\sim\sum_k B_k\hbar^{2k-1}t^{-s_k}/\Gamma(1-s_k)$, and of the resolvent-Mellin/analytic-continuation identity $Z(s)=\tfrac{1}{\pi}\sin(\pi s)M(s)$, where $M(s)$ is Parisi's Mellin transform of the resolvent trace of $H$ and $Z(s)=\mathrm{Tr}\,H^{-s}$ is the spectral zeta function, for the confining monomial class, with explicit remainder bounds, domains of validity in $s$, and uniformity in the semiclassical parameter (resolving the interplay between the $\hbar\to0$ limit and the analytic continuation in $s$), and establish the residue formula $Z_k=\tfrac{\sin(\pi s_k)}{\pi}B_k$ with $s_k=(\tfrac12+\tfrac1q)(1-2k)$ under those bounds. As part of this justification, the formal derivation of the coefficients $B_k$ (equivalently, the zeta residues $Z_k$) and their dependence on the WKB coefficients $C_i$ must be rigorously justified to arbitrary order for arbitrary even $q$. The proof may proceed by any route, but it must cover Parisi's identities as stated and not merely a special case. The identity-level statements are the short-time heat-trace expansion, the Mellin/continuation identity, and the residue formula; what part (1) newly requires is their quantitative control: explicit remainder bounds, domains of validity in $s$, uniformity in the semiclassical parameter, and rigorous arbitrary-order control of the coefficients $B_k$ (equivalently the residues $Z_k$). (2) Effective error control of the numerical scheme — prove convergence of, and give quantitative error bounds for, the truncation strategy that determines low-lying eigenvalues from convergent trace identities built by subtracting WKB approximants $W_n^{(j)}=\hbar^{2q/(q+2)}N^{2q/(q+2)}\sum_{i\le j}C_iN^{-2i}$ (with $N=n+\tfrac12$, $n$ the level index, and $C_i$ the WKB coefficients of the eigenvalue asymptotics) from the zeta sum: how the truncation order $j$, the number of retained exact eigenvalues, and the conditioning of the resulting nonlinear system control the achievable accuracy, for arbitrary even $q$ and target accuracies beyond the demonstrated $10^{-4}$ at $q=4$. Both deliverables are aspects of one objective — a rigorous and effective theory of Parisi's trace identities within the confining-monomial class.

References

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