数理解析研究所講究録 1424 巻 2005 年 214-231

214

The general 1D Schr\"odinger equation as anexactly solvable problem
Andre Voros *CEA, Service de Physiq ue Theorique de Saclay(CNRS URA 2306)
F-91191 Gif-sur-Yvette CEDEX, France
E-mail : voros@spht.saclay.cea.fr

December 13, 2004
We review an exact WKB resolution method for the stationary ID Schr6dinger equation with a general polynomial potential. This contribution covers
already published material: we supply a commented summary here, stressing
a few aspects which were less highlighted before. As some of our earlier
papers needed later corrections, we also recapitulate these here in footnotesin the bibliography. The latter, not meant to be exhaustive, focuses onnarrowly relevant works plus some broader ones that do offer more extensive
bibliographies.

1

General setting

We will treat a Schr\"odinger equation on the real line with a polynomial potential function(real, normalized as $V(q)=+q^{N}+v_{1}q^{N-1}+\cdots+v_{N-1}q$):$V$$(- \frac{\mathrm{d}^{2}}{\mathrm{d}q^{2}}+[V(q)+\lambda])\psi(q)=0$,(1)*Also at: Institut de Mathematiques de Jussieu-Chevaleret (CNRS UMR 7586), Universite Paris 7 F-75251 Paris CEDEX 05, Ikance.

215(a general Sturm-Liouville problem). It is convenient to pose the probiem on; exact treatments however rely on analyticala half-line, initially(wherecontinuations to the whole complex domain:$q\in[\mathrm{O}, +\infty)$$q\in \mathbb{C},$$(\prec v_{1}, .
$v\equiv$

1.1

.

.

, v_{N-1}))$$(\vec{v}, \lambda)\in \mathbb{C}^{N}$.

Outline of results [47]-[50]The traditional view about eq.(l) (in any dimension) is that
$N=2$ is exactly solvabie: the harmonic oscillator, “the nice” case;no other $N$ is solvable (generically) :$N=1$ gives the Airy equation (more transcendental than the higherdegree $N=2$ , aparadox!)$N\geq 3$ yield the anharmonic oscillators, the ground of choice for all
approximation methods (perturbative, semiclassical . . .).
--$\bullet$$\bullet$We see eq.(l) instead as an exactly solvable problem for any N,
thanks to an exact $WKB$ method with two key ingredients:-a semiclassical analysis exploiting zeta-regularization $[43, 29]$ ;-like but selfconexact quantization conditions that aresistent, and akin to the Bethe Ansatz equations that solve many exact
statistical mechanics $[21, 40]$ or quantum field theory [7].
models inMoreover, degrees $N$ and $4/N$ behave similarly in this framework (a qualitative duality) [46]:$N=1$ is transcendental iike $N=4$ , clarifying the above paradox;$N=2$ (selfdual) becomes completely explicitly solvable as a degenerate, rather pathoiogical, case. A countable farnily of similarly solvabte
cases is actually obtained [50, Sec.4]: all the generalized eigenvalue problems(for even degrees $N>0$ ).
$\mathrm{B}\mathrm{o}\mathrm{h}\mathrm{r}-\mathrm{S}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{e}\mathrm{i}\mathrm{d}$-$2\mathrm{D}$$\bullet$$\bullet$$[- \frac{\mathrm{d}^{2}}{\mathrm{d}q^{2}}+(q^{N}+\Lambda q^{\frac{N}{2}-1})]\psi(q)=0$

1.2

A few early sources

Exact asymptotics (which uses drastically divergent representations to perform fully exact calculations) is a paradoxical idea; its advent rests on var(restricting
ious seminal developments which mainly took place in theto works we felt as directly influential; a comprehensive bibliography would
need much more space):-Balian and Bloch’s representation of quantum mechanics, where the latter can be exactly rebuilt in principle just from the classicat trajectories,’ $70\mathrm{s}$216

provided these are taken in their full complex extension [5];the exact geometrical content of high-frequency asymptotics: singularities,
wave front sets, and their propagation properties [26, 15, 23]; (reviewed in[27, chap. VIII]-the same in the analytic (not) category (a key step to exactness) : ramified Cauchy problem [32], analytic pseudo-differential operators [11], hyperand micro-functions [36], localized Fourier transforms [13]; (reviewed in [39])Borel resummations of asymptotic series [20] , large-order perturbation theory [9], tunneling, instantons and Zinn-Justin’s conjectures-$)$$C^{\infty}$-$(\mathrm{e}\mathrm{v}\mathrm{g}., [25,6,33])$[53] .
$)$

seemingly unrelated direct monodromy calculations resulting in exact functional relations for Stokes multipliers [38].
The Balian-Bloch idea led to a first concrete exact WKB aigorithm for ID
problems like eq.(l), based on (generalized) Borel resummations [42, 30, 17,
18, 12, 28], and best formalized in Ecalle’s framework of resurgent functions[24]; the explicit exact WKB results are indeed special instances of bridge
equations in resurgence algebras (or resurgence equations). Still, in thisapproach, one basic resurgence theorem is not yet fully proved [1S, thm
1.2.1 and its Comment], and a fairly explicit solution stage has only beenreached for homogeneous polynomial potentials, rather tortuously [44]-[46].
Otherwise, many developments have been achieved in this and neighboring
directions: proofs of Bender-Wu $[30, 19]$ and Zinn-Justin conjectures $[17, 18]$ ,hyperasymptotics (reviewed in [28, Part ), results on Painleve’ functions(reviewed in [28, Part III]).
This presentation is about a more direct exact WKB method [47]-[50](surveyed in [51]), based upon functional relations \‘a la Sibuya [38], plus crucial inputs of zeta-regularization both at the quantum and classical levels.
At its core, it needs just one exact Wronskian identity plus a structure equation for a pair of spectral determinants, to generate the exact solution as
a fixed point of an explicit system of nonlinear equations (“quantization conditions”, but also Bethe-Ansatz equations in an integrable-modelinterpretation [21] . The Wronskian identity precisely subsumes the resurgence relations found in the Borel-transform approach, whereas the newer
exact quantization conditions (which achieve the solving task) have no clear
counterpart in the Borel plane as far as we know. This direct approach also
makes it easier to treat the general case at once, and it is now fully provenfor homogeneous potentials [4],-$\mathrm{I}\mathrm{I}]$$\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}$

$)$

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2

Basic ingredients

2.1

The conjugate equations [38]Just as a polynomial equation is better handted by treating all its conjugateroots together, the original differential equation (1) is to be supplemented byits conjugates, defined using$V^{[\ell]}(q)=\mathrm{e}^{-\mathrm{i}f\varphi}V(\mathrm{e}^{-\mathrm{i}\ell\varphi/2}q)\mathrm{d}\mathrm{e}\mathrm{f}$there are$L$,$\lambda^{[\ell]}=\mathrm{e}^{-\mathrm{i}\ell\varphi}\lambda \mathrm{d}\mathrm{e}\mathrm{f}$

with

,

distinct conjugate equations labeled by$L=\{$$N+2$$\frac{N}{2}+1$;$\varphi=\frac{4\pi}{N+2}\mathrm{d}\mathrm{e}\mathrm{f}$$\ell\in \mathbb{Z}/L\mathbb{Z}$(2), wheregenericaliy
for an even polynomial $V(q)$ .

(3)

Semiclassical asymptotics

2.2

WKB formulae for solutions of the Schr\"odinger equation [38]: a solution of eq.(l) which is recessive ( decaying) for $qarrow+\infty$ is specified by the
WKB behavior$\bullet$$=$$\psi_{\lambda}(q)\sim\Pi_{\lambda}(q)^{-1/2}\exp-\mathit{1}_{Q}^{q}.\Pi_{\lambda}(q’)\mathrm{d}q’$,$\Pi_{\lambda}(q)^{\mathrm{d}}=^{\mathrm{e}\mathrm{f}}(V(q)+\lambda)^{1/2}$,(4)

is theis the classical forbidden-region momentum (whereAgmon metric). Reexpansions in descending powers of $qarrow+\infty$ yield$\Pi_{\lambda}(q)\mathrm{d}q$$\Pi_{\lambda}(q)$$(5)$$[V(q)+ \lambda]^{-s-\vdash 1/2}\sim\sum_{\sigma}\beta_{\sigma}(s;\vec{v}, \lambda)q^{\sigma-Ns}$$( \sigma=\frac{N}{2}, \frac{N}{2}-1, \frac{N}{2}-2, \ldots),$(6)$\psi_{\lambda}(q)\sim \mathrm{e}^{C}q^{-N/4-\beta_{-1}(v^{\neg})}\exp\{-\sum_{\{\sigma>0\}}\beta_{\sigma-1}(\vec{v})\frac{q^{\sigma}}{\sigma}\}$when $N>2$ ).
(independent of A for
where, arises through the First
Another solution, recessive for
conjugate equation:$\sigma\geq 0$$\beta_{\sigma-1}(\vec{v})=\beta_{\sigma-1}(0;\vec{v}, \lambda)\mathrm{d}\mathrm{e}\mathrm{f}$$qarrow+\mathrm{e}^{-\mathrm{i}\varphi/2}\infty$$\Psi_{\lambda}(q)=\psi_{\lambda^{[1]}}(\mathrm{e}^{\acute{1}}q)\mathrm{d}\mathrm{e}\mathrm{f}[1]\varphi/2$(7).

and their g-derivatives
As the asymptotic form (6) and its analogs forhave overlapping sectors of validity, this explicit exact Wronskian folJows:$\Psi_{\lambda}(q)$$\Psi_{\lambda}(q)\psi_{\lambda}’(q)-\Psi_{\lambda}^{t}(q)\psi_{\lambda}(q)\equiv \mathrm{e}^{C+C^{[1]}}2\mathrm{i}\mathrm{e}^{\mathrm{i}\varphi/4}\mathrm{e}^{\mathrm{i}\varphi\beta_{-1}(v_{\grave{\mathit{1}}/2}^{\prec}}$$(\neq 0)$

.

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): the Schr\"odinger operator overSpectral asymptotics (withthe whole real line with the confining potential $V(|q|)$ has a purely discrete
spectrum. Due to parity-splitting,isthe Neumann, andthe Dirichlet, spectrum (the boundary
conditions being at $q=0$ ). As a corollary of the WKB formula (4) on eigenfunctions, the eigenvalues obey the semiclassical Bohr-Sommerfeld condition$E\equiv-\lambda$$\bullet$$\mathcal{E}^{+}\mathrm{d}\mathrm{e}\mathrm{f}=\{E_{2k}\}$$\{E_{k}\}_{k\in \mathrm{N}}(E_{k}\uparrow+\infty)$$\mathcal{E}^{-}\mathrm{d}\mathrm{e}\mathrm{f}=\{E_{2k+1}\}$for integer $karrow+\infty$ ,$S(E_{k}) \sim k+\frac{1}{2}$(9)in terms of the classicalactiion function
which be$Earrow+\infty$
haves asforwithFurther expansion of eq.(9) to all orders in
foilowed by reexpansion$E$in descending powers of
results in a complete asymptotic $(Earrow+\infty)$
eigenvalue formula, of the form [47]
$S(E)=^{\mathrm{d}\mathrm{e}\mathrm{f}} \oint_{\{p^{2}+V(q)=E\}}\frac{\mathrm{p}\mathrm{d}q}{2\pi},$$\mu=\frac{1}{2}+\frac{1}{N} (\mathrm{t}\mathrm{h}\mathrm{e} order).$$b_{\mu}E^{\mu}$$k^{-1}$$\sum_{\alpha}b_{\alpha}E_{k}^{\alpha}\sim k+\frac{1}{2}$for integer $karrow+\infty$(a $=\mu,$$\mu-\frac{1}{N},$$\mu-\frac{2}{N},$$\ldots$) (10)

are polynomial in the, and also parity-dependent but
not above $ce=-3/2$ nor if is an even polynomial).
(the$b_{\alpha}$$\{v_{j}\}_{j\leq(\mu-\alpha)N}$

$V$

2.3

Spectral functions [43]$\bullet$

Spectral zeta functions ($\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}arrow \mathrm{s}\mathrm{p}\mathrm{l}i\mathrm{t}$$({\rm Re} s>\mu)$$Z^{\pm}(s, \lambda)\mathrm{d}\mathrm{e}\mathrm{f}=\sum_{k_{\mathrm{o}\mathrm{d}\mathrm{d}}^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}}(E_{k}+\lambda)^{-s}$for later use):(and $Z(s)=Z^{+}(s)+Z^{-}(s)$ ),(11), regular at $s=0$ , all due toextend to meromorphic functions ofeq. (10).
Spectral determinants: formafly meant to be$s\in \mathbb{C}$$\bullet$$D^{\pm}( \lambda)="\prod_{k_{\mathrm{o}\mathrm{d}\mathrm{d}}^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}}(\lambda+E_{k})$

”,

(12)

they get rigorously specified through$\log D^{\pm}(\lambda)=^{\mathrm{d}\mathrm{e}\mathrm{f}}-\partial_{s}Z^{\pm}(s, \lambda)_{s=0}$(zeta-regularization),

(13)

21)

and further evaluated by limit formulae which we call structure equations(they are like Hadamard product formulae, but with no undetermined
constants whatsoever a crucial property here):$\log D^{\pm}(\lambda)=\lim_{K\prec+\infty}\{\sum_{k<K}\log(E_{k}+\lambda)+\frac{1}{2}\log(E_{K}+\lambda)$

(

$k,$

$K$

(14)

evenodd)(counterterms:)$- \frac{1}{2}\sum_{\{\alpha>0\}}b_{\alpha}(E_{K})^{\alpha}[\log E_{K}-\frac{1}{\alpha}]\}$):(andAs corollaries for, of order in ;is an entire function in$\log D$ has a large-A expansion of a severely constrained (““canonical”) form, or generalized Stirling expansion [29]:$D^{\pm}$

$D$

-

.

$D$

$\lambda$$(\lambda, \tilde{v})$$\mu$$\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}$-(degree 0)(degreesor(15)$\log D(\lambda)\sim[a_{1}\lambda(\log\lambda-1)]+a_{0}\log\lambda[+\sum_{\mu_{k}\not\in \mathrm{N}}a_{\mu k}\lambda^{\mu k}]$(degree 1)”$\mu_{k}\downarrow-\infty$)

where obviously no term can occur with degree $>\mu$ (growth restriction; e.g.,
as soon as $N>2$ ); but above all,, including additive constants
any extraneous power terms, are banned outright (canonical constraint).
, resp. (15), essentially restore classic results: theFor $N=2,$
analytical continuation by the Euler-Maclaurin formula, resp. the Stirling(Lerch formuia), withexpansion, for
the Hurwitz zeta function. Those two founding formulae of all asymptotics
involve zeta-regularization, as the Lerch formula shows; by inference, WKB
theory (a branch of asymptotics) also ought to be intimately tied to zetaregularization, but this aspect traditionally goes unnoticed.
$a_{1}\equiv 0$$b_{n}\lambda^{n}(n\in \mathrm{N})$$(b_{0}\lambda^{0})$$\mathrm{e}\mathrm{q}\mathrm{s}.(14)$$\zeta(s, \lambda)=$$-\partial_{s}\zeta(s, \lambda)_{s=0}\equiv\log\sqrt{27\Gamma}/\Gamma(\lambda)$

3

Fundamental exact identities

3.1

Canonical normalization of recessive solution(6) to be recessive for $qarrow+\infty$ , still
described by
The solution
awaits normalization. It proves useful to fix it also at $q=+\infty$ , as$\psi_{\lambda}(q)$$\mathrm{e}\mathrm{q}\mathrm{s}.(4),$$\psi_{\lambda}(q)\sim\Pi_{\lambda}(q)^{-1/2}\exp\int_{q}^{+\infty}\Pi_{\lambda}(q’)\mathrm{d}q’$$(qarrow+\infty)$

.

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( “the Agmon distance from to
However,then wish to define this “improper action integral” as$\int_{q}^{+\infty}\Pi_{\lambda}(q’)\mathrm{d}q^{l}$$+\infty$$q$”) diverges: we(17)$[ \int_{q}^{+\infty}(V(q’)+ \mathrm{A})^{1/2-s}\mathrm{d}q’]_{sarrow 0}$(analytical continuation from $\{{\rm Re} s>\mu\}$ to $s=0$ ), but $s=0$ is a singular(cf. eq. (6)). Now:
point in general: a simple pole of residue$\beta_{-1}(s;\vec{v})=0$ quite frequently (for any polynomial $V(q)$ of odd degree $N$ ;also for any even polynomial of degree $N=4M$ ; for any $V(q)=q^{N},$ $N\neq 2)$ ;then $s=0$ becomes a regular point and the prescription (17) is fine;, a stronger regularization is suggested by the quantumwhen
classical correspondence. “Classical spectral functions” are naturally definable, by mimicking the quantum formulae (It), (13), as$N^{-1}\beta_{-1}(\vec{v})$$\bullet$$\beta_{-1}(s;v)\prec\neq 0$$\bullet$(18)$Z_{\mathrm{c}1}(s, \lambda)\mathrm{d}\mathrm{e}\mathrm{f}=\int_{\mathbb{R}^{2}}\frac{\mathrm{d}q\mathrm{d}p}{2\pi}(p^{2}+V(|q|)+\lambda)^{-s}$$\log D_{\mathrm{c}1}(\lambda)\mathrm{d}\mathrm{e}\mathrm{f}=-\partial_{s}Z_{\mathrm{c}1}(s, \lambda)_{s=0}$

.

(19)

These formulae together imply a classical counterpart to the formal eq.(12),” $\int_{-\infty}^{+\infty}(V(|q|)+\lambda)^{1/2}\mathrm{d}q$$”=\log D_{\mathrm{c}1}(\lambda)$

(20)

;

using parity symmetry, this further splits to give the pair [50, Sec.1.2.2]$D_{\mathrm{c}1}^{\pm}(\lambda)=\Pi\lambda(0)^{\pm 1/2}\exp$$\log D_{\mathrm{c}1}^{\pm}(\lambda)$$\log D^{\pm}(\lambda)$“ $f_{0}^{+\infty}\Pi_{\lambda}(q)\mathrm{d}q$

”;

(21)

are also the divergent parts of the large-A expansions (i5) for, so they are canonical as well. The idea is then to define

to be
(and
solution, with a normalization
would turn up elsewhere if not here):$\frac{1}{2}\log D_{\mathrm{c}1}(\lambda)$$\int_{0}^{+\infty}\Pi_{\lambda}(q)\mathrm{d}q$). This in turn fixes the recessivein eq.(6) (an anomaly factor, which$I_{q}^{+\infty}= \int_{0}^{+\infty}-\int_{0}^{q}$$\mathrm{e}^{\mathrm{C}}\not\equiv 1$$C \equiv\frac{1}{N}[-(2\log 2)\beta_{-1}(s,\cdot\vec{v})+\partial_{s}(\frac{\beta_{-1}(s\vec{v})}{1-2s})]_{s=0}$

.

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As examples of the resulting im proper action integrals: [52]$\int_{0}^{+\infty}(q^{N}+\lambda)^{1/2}\mathrm{d}q=-\frac{1}{2}\pi^{-1/2}\Gamma(\frac{1}{2}+\mu)\Gamma(-\mu)\lambda^{\mu}$$\int_{0}^{+\infty}(q^{2}+\lambda)^{1/2}\mathrm{d}q=-\frac{1}{4}\lambda(\log\lambda-1)$$(N \neq 2; \mu=\frac{1}{2}+\frac{1}{N})$(anomalous when \lambda$\neq 0$)$\int_{0}^{+\infty}(q^{4}+vq^{2})^{1/2}\mathrm{d}q=..-\frac{1}{3}v^{3/2}$

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Remarks:(or) is not an entire function of A like :-a “classical determinant”
the discrete spectrum of zeros of classically becomes a continuous branch.
cut for-our treatment of the normalization for
was initially fiawed [47, 48]:
prejudged
thenaturalanswertoremain $C=0$ , so we had temwe erroneously
porary inconsistencies of normalization in the anomalous case; these affecteqs.(15-19) (26) in [47] and Secs.l.l, 2.1, 2.2 in [50], but cancel themselves
spontaneously thereafter; the error is repaired from [49] onwards.
$D$$D_{\mathrm{c}1}^{\pm}$$D_{\mathrm{c}1}$$D$$D_{\mathrm{c}1}$$\beta_{-1}\not\equiv 0$

$,$

3.2

Basic exact identities

Under the canonicai normalization for the recessive solution, and with$D^{+}(\lambda)\equiv-\psi_{\lambda}’(0)$

resp.

,

$D^{-}(\lambda)\equiv\psi_{\lambda}(0)$$’= \frac{\mathrm{d}}{\mathrm{d}q}$

.

,

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The idea, assuming $N>2$ , is first to prove the logarithmic A-derivatives of
eqs.(24) by anatytically computing traces of the resolvent kernel, and tointegrate back: then, no integration constants can reappear precisely because
all four logarithms have the canonical targe-A behavior (15), in particular$\lambda-$$\psi_{\lambda}’(0)\sim-D_{\mathrm{c}1}^{+}(\lambda)$

,

resp.

$\psi_{\lambda}(0)\sim D_{\mathrm{c}1}^{-}(\lambda)$(25)). In depth, eqs.(24)
and (16) at $q=0$ , also valid for A
exception,from eq.(i4)constants
without
arose by controlling all integration
onwards.
The basic exact identities (24) can now be substituted into the explicitly
known Wronskian (8) computed at $q=0$ , resuiting in a functional relation
betvveen spectral determinants or “Wronskian identity” , also fundamentai:(from$\mathrm{e}\mathrm{q}\mathrm{s}.(21)$$arrow+\infty$$\mathrm{e}^{+\mathrm{i}\varphi/4}D^{[1]+}(\mathrm{e}^{-\mathrm{i}\varphi}\lambda)D^{-}(\lambda)-\mathrm{e}^{-\mathrm{i}\varphi/4}D^{+}(\lambda)D^{[1]-}(\mathrm{e}^{-\mathrm{i}\varphi}\lambda)\equiv 2\mathrm{i}\mathrm{e}^{+\mathrm{i}\varphi\beta_{-1}(\vec{v})/2}$

4

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Exact quantization conditions

At first glance, the exact functional relations (26) look grossly underdetermined; like the original Wronskian formula (8), they amount to one equation
for two unknown functions. The exact information here seems to present a“missing link” throughout. This is however a delusion, because those unknown functions have a truly special form here, embodied in the structure

222

equations (14) (themselves consequences of zeta-regularization), and this precisely fills that information gap.

Degenerate cases

4.1$N=2$ :

(harmonic oscillator).
, and it has A-dependence.
This is the most singular case:Now,makes eq.(26) degenerate: for A real, it separates into real andimaginary parts, which recombine as(A-1), Thezeros are then obvious, and they clearly split betweenoraccording to their sign; thereupon, the structure equations (14) yield the
explicit solutions$\bullet$$[- \frac{\mathrm{d}^{2}}{\mathrm{d}q^{2}}+(q^{2}+\lambda)]\psi(q)=0$$\beta_{-1}=\lambda/2\not\equiv 0$$\varphi=\pi$$D^{+}( \lambda)D^{-}(-\lambda)=2\cos\frac{\pi}{4}$$D^{-}(-\lambda)$$D^{+}(\lambda)$$D^{+}( \lambda)=\frac{2^{-\lambda/2}2\sqrt{\pi}}{\Gamma(\frac{1+\lambda}{4})}$,$D^{-}( \lambda)=\frac{2^{-\lambda/2_{\sqrt{\pi}}}}{\Gamma(\frac{3+\lambda}{4})}$

even: the generalized eigenvalue probtems [

.

(27)

)] $\psi(q)=0$
A
are also exactiy solvable [50, Sec.4]: settingand (14) likewise lead to the relevant Neumann and Dirichlet determinants for $q\in[0, +\infty)$ ,$\bullet$$N$$- \frac{\mathrm{d}^{)}\lrcorner}{\mathrm{d}q^{2}}+(q^{N}+$$\nu=\frac{1}{N+2},$$q^{\frac{N}{2}-1}$$\mathrm{e}\mathrm{q}\mathrm{s}.(26)$as$D_{N}^{+}( \Lambda)=-\frac{2^{-\Lambda/N}(4\iota/)^{\nu(\Lambda+1)+1/2}\Gamma(-2\nu)}{\Gamma(\nu(\Lambda-1)+1/2)}$,$D_{N}^{-}( \Lambda)=\frac{2^{-\Lambda/N}(4\nu)^{\iota/(\Lambda-1)[perp] 1/2}\Gamma(2\nu)}{\Gamma(\nu(\Lambda+1)+1/2)}$(28)

whereas over the whole real q-axis,$D_{N}^{+}(\Lambda)D_{N}^{-}(\Lambda)$$\det[-\frac{\mathrm{d}^{2}}{\mathrm{d}q^{\mathit{2}}}.+(q^{N}+\Lambda q^{\frac{N}{2}-1})]=\{$$(\sin\pi\nu)^{-1}\cos\pi\nu\Lambda$

if
if $N\equiv 0$$N\equiv 2 (\mathrm{m}\mathrm{o}\mathrm{d} 4)$$(\mathrm{m}\mathrm{o}\mathrm{d} 4)$.

(29)
Remark: actually, the potentials
supersymmetricareat thegeneralized eigenvalues A (see [50] and the footnote under that reference).
$(q^{N}+\Lambda q^{\frac{N}{2}-1})$

4.2

The generic situation

The preceding treatment does not generalize straightforwardly. Instead, themere division of the functional relation (26) at $\lambda=-E_{k}$ by its first conjugate
partner leads to the pair of formulae$2 \arg D^{[1]\pm}(-\mathrm{e}^{-\mathrm{i}\varphi}E_{k})-\varphi\beta_{-1}(\vec{v})=\pi[k+\frac{1}{2}\pm\frac{N-2}{2(N+2)}]$for$k_{\mathrm{o}\mathrm{d}\mathrm{d}}^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}$, (30);

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which have the outer form of Bohr-Sommerfeld quantization conditions but
are exact, albeit implicit (the left-hand sides invoke the determinants of thefirst conjugate spectra, equally unknown).
However, we can write the totality of such conditions for the conjugate
problems (Sec.2.1), together with the structure equations that express all theand $K$ evenodd’
determinants from their spectra: namely, for
$L$

$k,$

$\ell\in \mathbb{Z}/L\mathbb{Z}$$\mathrm{i}^{-1}[\log D^{[l+1]\pm}(-\mathrm{e}^{-\mathrm{i}\varphi}E_{k}^{[l]})-\log D^{[l-1]\pm}(-\mathrm{e}^{+\mathrm{i}\varphi}E_{k}^{[\ell]})]-(-1)^{l}\varphi\beta_{-1}(\vec{v})$(31)$= \pi[k+\frac{1}{2}\pm\frac{N-2}{2(N+2)}]$$\log D^{[\ell]\pm}(\lambda)\equiv\lim_{Karrow+\infty}\{$

,

(32)

$\sum_{k<K}\log(E_{k}^{[\ell]}+\lambda)+\frac{1}{2}\log(E_{K}^{[f]}+\lambda)$$- \frac{1}{2}\sum_{\{\alpha>0\}}b_{\alpha}^{[f]}(E_{K}^{[\ell]})^{\alpha}[\log E_{K}^{[\ell]}-\frac{1}{\alpha}]\}$.

As a result, the exact spectra now appear as solutions of nonlinear, paritysplit, selfconsistent or Fixed-point equations of the form$\mathrm{A}t^{+}\{\mathcal{E}^{\cdot}+\}=\mathcal{E}^{\cdot}+$

,

resp.

(33)

$\lambda 4^{-}\{\mathcal{E}^{\cdot}-\}=\mathcal{E}^{\cdot}-)$.
denote the disjoint unions of the conjugate spectra
where$N\leq6$
and not too big coefficients , suggest
Numerical tests, done for(31)-(32)can be cast into contractive form.
equations
fixed-pointthat theAssuming this conjecture, the problems (33) get solved simply by forward
iterations, which induce convergence tow ard the exact spectra starting fromtrial spectra that need only be semiclassically correct (to a certain order).
$\mathcal{E}^{[f]\pm}$$\mathcal{E}^{\cdot}\pm$$\vec{v}$4.3

Example: homogeneous potentials $q^{N}[44]-[46]$This is the simplest case (all conjugate spectra coincide, and A is the only
variable), and specially so if $N>2$ (see Sec.4.1 for $N=2,$ $[46]$ for $N=1$ ).
For $N>2$ , the exact quantization conditions (31) reduce to a single pair(e.g., see fig.l for $N=4$ ),2$\arg D^{\pm}(-\mathrm{e}^{-\mathrm{i}\varphi}E_{k})=\pi[k+\frac{1}{2}\pm\frac{N-2}{2(N+2)} ]$for$k=_{1,3,5\ldots}^{0,2,4,\ldots}$(34)

while the structure equations (32) simplify as$(E>0))$$\arg D^{\pm}(-\mathrm{e}^{-\mathrm{i}\varphi}E)\equiv\sum_{k_{\mathrm{o}\mathrm{d}\mathrm{d}}^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}}\arg(E_{k}-\mathrm{e}^{-\mathrm{i}\varphi}E)$

(35)

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Figure 1: The exact quantization formulae (34) ( , left; -, right) for the homogeneous quartic potentiai : the curves(solid lines labeled by-values) cross the (dashed) lines of prescribed argument in the complex D-plane.
$+$$\{D^{\pm}\langle \mathrm{e}^{\mathrm{i}\pi/3}E)\}_{E>0}$$q^{4}$$E$$\kappa_{41}=(N-2)/(N+2)$

Figure 2: Numericaliy measured linear contraction factors

at the fixed points(i.e., the exact -Ax spectra), upon iterations of the exact quantization conditions(34)-(35) for homogeneous potentials. As horizontal scale we use the value
taken byclassically (whenis replaced byas in eq. (21)):.
$\hslash\pm$$|q|^{N}$$D^{\pm}$$\kappa_{\pm}$$D_{\mathrm{c}1}^{\pm}$$\kappa_{\mathrm{c}1}(N)\equiv\frac{N-2}{N+2}$

225

for $karrow+\infty$ .
and the asymptotic boundary condition is
The latter fixed-point equations have now been proven globally contractive on suitable sequence spaces [4], by methods specific to the homogeneouscase (see fig.2). (For the more generai equations (31)-(32), contractivity
remains a conjecture.)
Those very same equations (34)-(35) have also been found to yield thestatistical mechanics
ground states of some exactly solvable models in$[21, 40]$ and in conformal quantum field theories [7] (Bethe Ansatz equations).
A deep explanation for this coincidence of equations (and, by uniqueness, of
their solutions) in so different dynamical settings is not yet available. This“
correspondence” (as a dictionary between Schr\"odinger Ordinary
Differential Equations and integrable Models) currently applies to homogeneous problems (even singular ones) plus the supersymmetric equations of
Sec.4.1 on the Schr\"odinger side, and onty to the ground (non-excited) statesin integrable models. There is now no integrable-model realization of the general inhomogeneous quantization conditions (31)-(32); by extension, we still
consider these as Bethe Ansatz equations, and the general ID Schr\"odinger
equation (1) as exactly soivable through them.
$b_{\mu}E_{k}^{\mu} \sim k+\frac{1}{2}$$2\mathrm{D}$$\mathrm{O}\mathrm{D}\mathrm{E}/\mathrm{I}\mathrm{M}$5

Extensions: realized vs desirable

5.1

Solving for the unknown functions $\psi[47]-[51]$of eq.(l) as an example. By
We keep focusing on the recessive solution
Jetting now the reference half-line vary as $[Q, +\infty)$ , we obtain parametric,translates of the basic identities (24), essentiallythus amounts to a parametric spectral. The computation ofprobiem, and this is now exactly solvable through parametric fixed-point. (This has also underfor
equations like (33), e.g.,gone successful numerical tests.)$\psi_{\lambda}$$\psi_{\lambda}(Q)\equiv D_{Q}^{-}(\lambda)$$-D_{Q}^{+}(\lambda)$$\psi_{\lambda}(Q)$$\mathcal{M}_{Q}^{-}\{\mathcal{E}_{Q}^{-}.\}=\mathcal{E}_{Q}^{-}.$

5.2$\psi_{\lambda}’(Q)\equiv$$\psi_{\lambda}(Q)$

Toward quantum perturbation theory [8, 52]

By rescalings \‘a la Symanzik, the perturbative regime for an anharmonic potential like $V(q)=q^{2}+gq^{4}$ amounts to the $varrow+\infty$ regime for the potential$q^{4}+vq^{2}$ .
This limit is not manifest, and is actually singular, in our exact
quantization formalism. We recently obtained explicit large-v behaviors for226such spectral determinants, basicalJy as$\frac{\det^{\pm}(-\frac{\mathrm{d}^{2}}{\mathrm{d}q^{2}}+q^{N}+vq^{M}+\lambda)}{\det^{\pm}(-\frac{\mathrm{d}^{2}}{\mathrm{d}q^{2}}+vq^{M}+\lambda)}\sim\frac{\det_{\mathrm{c}1}^{\pm}(-\frac{\mathrm{d}^{2}}{\mathrm{d}q^{2}}+q^{N}+vq^{M}+\lambda)}{\det_{\mathrm{c}1}^{\pm}(-\frac{\mathrm{d}^{2}}{\mathrm{d}q^{2}}+vq^{M}+\lambda)}$(36)(for $varrow+\infty$ , with $N>M$ ); i.e., the singular (divergent) large-p behavior
is entirely concentrated in the classical parts of the determinants; these now
express in terms of improper action integrals (cf. Sec.3.1),$\frac{\det_{\mathrm{c}1}^{\pm}(-\frac{\mathrm{d}^{2}}{\mathrm{d}q^{2}}+q^{N}+vq^{M}+\lambda)}{\det_{\mathrm{c}1}^{\pm}(-\frac{\mathrm{d}^{2}}{\mathrm{d}q^{2}}+vq^{M}+\lambda)}\equiv\frac{\exp\int_{0}^{+\infty}(q^{N}+vq^{M}+\lambda)^{1/2}\mathrm{d}q}{\exp\int_{0}^{+\infty}(vq^{M}+\lambda)^{1/2}\mathrm{d}q}$

,

(37)

and the latter can be estimated in the end [52].

5.3

Generalizations?

It is most desirabie to push exact solvability beyond polynomiat potentials.
According to some exact results, rational potentials $[2, 7]$ and trigonometricpolynomials $[53, 16]$ appear as realistic targets, as well as some differential equations of order $>2[1,41,22,3]$ . The general Heun class is also a
possible direction [31], which includes other important equations (Mathieu,
Lam\’e, , . . ).
Of course, an even greater challenge is to tackle higher-dimensional Schr6dinger-type problems, in line with the original Balian-Bloch surmise itself[5]. Exact asymptotic methods are still in the exploratory stage there, seefor instance [14, 35, 10, 37, 34].
Therefore the subject is far from being closed!

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for [50]: 1) the term “quasi-exactly solvable” was wrongly put for “su-$4\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$persymmetric” throughout (without practical consequence; the two notions happen to;2) regarding the Airy zeros in Table 1,agree when $N=2$ and 6, but not$Z_{1}^{+}’(0)\approx 0.0861126,$
1.2585417.
,
0.9174909$\mathrm{b}\mathrm{e}\mathrm{y}\mathrm{o}\mathrm{n}\mathrm{d}\backslash ,$$\mathrm{e}^{-Z_{1}^{+\prime}(0)}\approx$$\mathrm{e}^{-Z_{1}^{-\prime}(0)}\approx$