Generalized ODE/IM Correspondence

Katsushi Ito (Tokyo Institute of Technology)
September 14, 2019
Workshop New Trends in Integrable Systems 2019@Osaka City University
arXiv:1811.04812[hep-th], JHEP 01 (2019) 228
in collaboration with Marcos Mariño, Hongfei Shu

1

Introduction

Introduction
We consider the second order diﬀerential equation:−ℏ2d2
ψ(q) + (V (q) − E)ψ(q) = 0.
dq 2

where V (q) is a polynomial in q.
This ODE appears in• quantum mechanics: 1D Schrödinger equation• 4d SUSY gauge theory [Mironov-Morozov]
quantum SW curve in N = 2 gauge theory(ℏ: Omega deformation parameter in the Nekrasov-Shatashvili
limit)
2

Quantum mechanics and resurgenceexact WKB method [Voros 1981, Sato-Aoki-Kawai-Takei, ... ]• the WKB expansion is asymptotic expansion in ℏ
• Borel and Laplace transformations
ϕ(z) =∞
X
n=0

cn z −n−1 → ϕ̃(ξ) =

∞
X

cn

n=0

→ L[ϕ̃](z) =

ξn
n!

Z ∞

dξe−ξz ϕ̃(ξ)

0• resurgence (perturbative ←→ non-perturbative) [Ecalle]• the exact WKB periods show discontinuity across the Stokes
lines
Discontinuity formula [Delabaere-Dillinger-Pham 1993]
3

Analytic bootstrap• Voros’ idea: The classical periods and their discontinuity
structure determine the exact WKB periods.
analytic bootstrap or the Riemann-Hilbert problem• The exact WKB periods and the exact quantization condition
determine the exact spectrum of QM.
• explicitly worked out for cubic and quartic potentials

4ODE/IM correspondence

The ODE/IM correspondence [Dorey-Tateo 9812211]• a relation between spectral analysis approach of ordinary
diﬀerential equation (ODE), and the “functional relations”
approach to 2d quantum integrable model (IM).
• Stokes coeﬃcients of the solutions satisfy functional relations• Baxter’s T-Q relation, T-system, Y-system=⇒ NLIE, Thermodynamic Bethe Ansatz (TBA) equations• TBA equations solve the spectral determinants and the exact
WKB periods of QM with the monic potential V (x) = x2M .

5

KI-Mariño-Shu 2018

We present the TBA equations governing the exact WKB periods
for general polynomial potential. These TBAs also provide ageneralization of the ODE/IM correspondence.
We will discuss• the relation between the discontinuity formula in Quantum
Mechanics and the TBA equations• generalization of the ODE/IM correspondence
for general polynomial potential.
We then apply the TBA equations to solve the spectral problem in
QM.
6Introduction

Exact WKB and resurgent quantum mechanics

Generalized ODE/IM correspondence

Example: cubic potential

Conclusions and outlook

7

Exact WKB and resurgent quantum
mechanics

Exact WKB method
the Schrödinger equation−ℏ2 ψ ′′ (q) + (V (q) − E)ψ(q) = 0.
the WKB solution:

 Z q

i
′
′
ψ(q) = exp
Q(q )dq .
ℏQ2 (q) − iℏ
Q(q) =∞XdQ(q)= p2 (q),dqp(q) = (E − V (q))1/2 ,Qk (q)ℏk = Qeven + Qodd = P (q) +k=0

P (q) = Qeven =X

pn (q)ℏ2n ,iℏ dlog P (q).
2 dq

Qodd = total derivativen≥0

p0 (q) = p(q) and pn (q) are determined recursively.
8

WKB periods and Voros symbols
potential: polynomial in q
V (q) = q r+1 + u1 q r + · · · + ur q
the WKB curve: hyperelliptic Riemann surface of genus g = r+1 2

ΣWKB : y = 2(E − V (q)).
2

WKB periods (quantum periods)IΠγ (ℏ) = P (q)dq,

γ ∈ H1 (ΣWKB ).

γΠγ (ℏ) =

X

I

2n
Π(n)
γ ℏ ,

Π(n)
γ =

pn (q)dq.
γn≥0Voros symbolVγ = exp


i
Πγ .
ℏ
9

Borel resummation

The WKB expansion of the quantum periods:
IX(n) 2n(n)Πγ (ℏ) =Πγ ℏ ,
Πγ = pn (q)dq.
γ

n≥0

(n)

• asymptotic expansion: Πγ ∼ (2n)!.
• The Borel transformationb γ (ξ) =
Π

X
n≥0

1
Π(n) ξ 2n ,(2n)! γ10Laplace transformation

The Borel resummation (Laplace transformation)Z
1 ∞ −ξ/ℏ be
Πγ (ξ)dξ,ℏ ∈ R>0 .
s (Πγ ) (ℏ) =ℏ 0Borel resummation along a direction speciﬁed by an angle φ:sφ (Πγ ) (ℏ) =1ℏ

Z eiφ ∞b γ (ξ)dξ.
e−ξ/ℏ Π011

lateral Laplace tranformation• singularity in the φ-direction in the ξ-plane• the lateral Borel resummations along a direction φ:
sφ± (Πγ ) eiφ ℏ = lim s (Πγ ) eiφ±iδ ℏ ,ℏ ∈ R>0 ,δ→0+• The discontinuity across the direction φ
discφ (Πγ ) = sφ+ (Πγ ) − sφ− (Πγ ).
4

2

-6

-4

2

-2

4

6

-2

-4

cubic potential
12Delabaere-Pham formula
WKB periods• cycle γ1 : a classically allowed interval E > V (x)• cycle γ2 : a classically forbidden interval E < V (x)<latexit sha1_base64="q8ezL9vhs+8lX9b632ackECwrMk=">AAAB73icbZBNSwMxEIZn61etX1WPXoJF8FR2RdCTFLx4rGA/oF3KbJptQ5PsmmSFUvonvHhQxKt/x5v/xrTdg7a+EHh4Z4bMvFEquLG+/+0V1tY3NreK26Wd3b39g/LhUdMkmaasQROR6HaEhgmuWMNyK1g71QxlJFgrGt3O6q0npg1P1IMdpyyUOFA85hSts9rdAUqJvaBXrvhVfy6yCkEOFchV75W/uv2EZpIpSwUa0wn81IYT1JZTwaalbmZYinSEA9ZxqFAyE07m+07JmXP6JE60e8qSuft7YoLSmLGMXKdEOzTLtZn5X62T2fg6nHCVZpYpuvgozgSxCZkdT/pcM2rF2AFSzd2uhA5RI7UuopILIVg+eRWaF9XA8f1lpXaTx1GEEziFcwjgCmpwB3VoAAUBz/AKb96j9+K9ex+L1oKXzxzDH3mfP68Dj7U=</latexit>

1<latexit sha1_base64="22Nw1JyUfXG60VpOTTbQseQk/Sg=">AAAB73icbZBNSwMxEIZn/az1q+rRS7AInspuEfQkBS8eK9gPaJcym2bb0CS7JlmhlP4JLx4U8erf8ea/MW33oK0vBB7emSEzb5QKbqzvf3tr6xubW9uFneLu3v7BYenouGmSTFPWoIlIdDtCwwRXrGG5FaydaoYyEqwVjW5n9dYT04Yn6sGOUxZKHCgec4rWWe3uAKXEXrVXKvsVfy6yCkEOZchV75W+uv2EZpIpSwUa0wn81IYT1JZTwabFbmZYinSEA9ZxqFAyE07m+07JuXP6JE60e8qSuft7YoLSmLGMXKdEOzTLtZn5X62T2fg6nHCVZpYpuvgozgSxCZkdT/pcM2rF2AFSzd2uhA5RI7UuoqILIVg+eRWa1Urg+P6yXLvJ4yjAKZzBBQRwBTW4gzo0gIKAZ3iFN+/Re/HevY9F65qXz5zAH3mfP7CHj7Y=</latexit>2• Πγ1 : discontinuous along the real positive axis
• Πγ2 : continuous along the real positive axis





i
discφ=0 (Πγ1 ) (ℏ) = −iℏ log 1 + exp − Πγ2 (ℏ).
ℏ[Delabaere-Pham 1999, Iwaki-Nakanishi]
13

Exact quantization condition

Bohr-Sommerfeld quantization condition1s(Πp )(ℏ) = 2πℏ n +, n = 0, 1, 2, . . .
2=⇒ E = En (ℏ) : perturbative spectrum
Exact quantization condition [V, DP, Zinn-Justin,..., Grassi-Marino]
2 cos(11s(Πp )(ℏ)) + e− 2ℏ s(Πnp )(ℏ) = 0
2ℏ(cubic potential)=⇒ E = En (ℏ) : energy spectrum
1ℏn = xn (E): Voros spectrum

14

Generalized ODE/IM
correspondence

TBA equations from DP formula• Vr+1 (q): a polynomial potential of degree r + 1• all the turning points qi , i = 1, · · · , r + 1, are real anddiﬀerent (q1 < q2 < · · · < qr+1 .)• γ2i−1 : [q2i , q2i ] classically allowed
γ2i : [q2i , q2i+1 ] classically forbidden• period integrals
Z q2i

m2i−1 = Π(0)γ2i−1 = 2q2i−1

p(q)dq, m2i = i Π(0)γ2i = 2i

Z q2i+1p(q)dq,q2i

are real and positive.

E<latexit sha1_base64="iyKsJ7k+bPLi33qq8l5aoG6aFEk=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSERRPBRE8tmA/oA1ls520azebsLsRSugv8OJBEa/+JG/+G7dtDtr6wsLDOzPszBskgmvjut9OYW19Y3OruF3a2d3bPygfHrV0nCqGTRaLWHUCqlFwiU3DjcBOopBGgcB2ML6d1dtPqDSP5YOZJOhHdCh5yBk11mrc9csVt+rORVbBy6ECuer98ldvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQuxYljVD72XzRKTmzzoCEsbJPGjJ3f09kNNJ6EgW2M6JmpJdrM/O/Wjc14bWfcZmkBiVbfBSmgpiYzK4mA66QGTGxQJnidlfCRlRRZmw2JRuCt3zyKrQuqp7lxmWldpPHUYQTOIVz8OAKanAPdWgCA4RneIU359F5cd6dj0VrwclnjuGPnM8floeMwQ==</latexit><latexit···

15

TBA equations• discontinuity of Πγ2i −1 is determined by the DP formula.
idisc Πγ2i−1 = −iℏ log 1 + exp − Πγ2i−2 (ℏ)
ℏ



i
− iℏ log 1 + exp − Πγ2i (ℏ),ℏ• discontinuity of Πγ2i : ℏ → ±iℏ (E − V (x) → V (x) − E)
Introduce the spectral parameter θ = − log ℏ (eθ = ℏ1 )




iπ
1
−iϵ2i−1 θ +± iδ = s± Πγ2i−1 (ℏ),
2ℏ−iϵ2i (θ) =1s (Πγ2i ) (ℏ),ℏ

The DP-formula reads
disc π2 ϵa (θ) = La−1 (θ) + La+1 (θ),
a = 1, · · · , r,
La (θ) = log 1 + e−ϵa (θ) ,L0 = Lr+1 = 0

16

Discontinuity and integral equations
Integral transformation
Z ∞Z
θ′ − θ
dx x + ζ′
F (ζ) =
f (x) =dθ′ cothf (eθ )
x x−ζ
2
0
R
′

ζ = eθ , x = eθThe discontinuity of F (ζ) along the positive real axis
Z ∞
dx
4iϵx
F (ζ + iϵ) − F (ζ − iϵ) =f (x) → 2πif (ζ)x(x−ζ)2 + ϵ20
discontinuity at φ = π/2 and −π/2 =⇒ TBA equationsZ ∞Z ∞La−1 (θ′ ) dθ′
La+1 (θ′ ) dθ′ϵa (θ) = ma eθ −−,′′−∞ cosh(θ − θ ) 2π
∞ cosh(θ − θ ) 2π
asymptotics ϵa (θ) → ma eθ (θ → ∞)17

generalized ODE/IM correspondence
ODE−∂z2 + z r+1 +rX!
ba z r−aψ(z, ba ) = 0

z∈C

a=1

• invariant under the rotation (Symanzik rotation)(z, ba ) → (ωz, ω a+1 ba ),2πi

ω = e r+3 .

• asymptotically decaying solution along the positive real axis


r+3
2
1
z 2 ,
y(z, ba ) ∼ √ z nr exp −r+32i
Stokesnsector
o
π
2kπ
< r+3
Sk = z ∈ C : arg(z) − r+3.
decaying solutions in Sk :yk (z, ba ) = ω 2 y(ω −k z, ω −(a+1)k ba ).
k18

Y-functions and Wronskian
WronskianWk1 ,k2 (ba ) ≡ yk1 (z, ba )∂z yk2 (z, ba ) − yk2 (z, ba )∂z yk1 (z, ba )• independent of z, W0,1 (= 1).
• periodicity[2]

Wk1 +1,k2 +1 (ba ) = Wk1 ,k2 (ba ).
f [j] (z, ba ) := f (ω −j/2 z, ω −j(a+1)/2 ba ).
Y-function
W−j,j (ba )W−j−1,j+1 (ba ),W−j−1,−j (ba )Wj,j+1 (ba )W−j−1,j (ba )W−j−2,j+1 (ba ) [+1],Y2j+1 (ba ) =W−j−2,−j−1 (ba )Wj,j+1 (ba )Y2j (ba ) =19

Y-system
Y-system (← Plücker identities for 2 × 2 determinants)Ys[+1] (ba )Ys[−1] (ba ) = 1 + Ys−1 (ba ) 1 + Ys+1 (ba ) .
boundary conditions: Y0 = Yr+1 = 0
Ar -type Y-system• b1 = · · · = br−1 = 0, br = E (monomial pot. ) [Dorey-Tateo]• multiple spectral parameters b1 , · · · , br
• It is diﬃcult to write down TBA equations
6-term TBA for cubic potential Masoero 1005.1046
NLIE for x6 + αx2 -potential Suzuki 0003066
20generalized ODE/IM
Introduce scaled variables2

q = ζ r+3 z,

2(a+1)

ua = −ζ r+3 ba ,−ζ 2 ∂q2 + q r+1 −rX

a = 1, 2, · · · , r.
!

ua q

r−a

ψ̂(x, ua , ζ) = 0.

a=1

• regard ζ as a new spectral parameter (ζ = ℏ in QM)• Symanzik rotation ↔ rotation of ζ2(a+1)ŷ(q, ua , ζ) = y(z, ba ) = y(ζ − r+3 q, −ζ − r+3 ua ).

2(a+1)
2−k−(a+1)kiπk − r+3iπk − r+3
(ω z, ωba ) = (e ζ)q, −(e ζ)ua .
2• subdominant solution in the sector Skkŷk (q, ua , ζ) = ω 2 ŷ(q, ua , eiπk ζ).

21

Y-system
Wronskian2Ŵk1 ,k2 (ζ, ua ) = ζ r+3 ŷk1 ∂q ŷk2 − ŷk2 ∂q ŷk1 (q, ua , ζ) = Wk1 ,k2 (ba ).
Y functions
Y2j (ζ, ua ) =πi

Y2j+1 (e− 2 ζ, ua ) =Ŵ−j,j Ŵ−j−1,j+1(ζ, ua ),Ŵ−j−1,−j Ŵj,j+1
Ŵ−j−1,j Ŵ−j−2,j+1(ζ, ua ).
Ŵ−j−2,−j−1 Ŵj,j+1

Y-system:




πi
πi
Ys (ζe 2 , ua )Ys (ζe− 2 , ua ) = 1 + Ys+1 (ζ, ua ) 1 + Ys−1 (ζ, ua ) ,• ua ﬁxed, single spectral parameter ζ• massless limit of Y-system in AdS3 minimal surface withlight-like boundary [Alday-Maldacena-Sever-Vieira, Hatsuda-KI-Sakai-Satoh]22

TBA equation
asymptotics:



Z
iδk q
′
′
P (q )dq .
ŷk (q, ua , ζ) ∼ (−1) c(ζ) exp −ζ q(k)
k2

asymptotic behavior of the Y-function ζ → ∞
I
1
mr−2k
log Y2k+1 (ζ, ua ) ∼ −p(q)dq = −,ζ γr−2kζIimr+1−2k ,log Y2k (ζ, ua ) ∼ −p(q)dq = −,ζ γr+1−2kζζ = e−θ ,Ya (ζ) = e−ϵa (θ)

TBA equations (⇐= Y-system+asymptotics [Al. Zamolodchikov] )Z
Z
La−1 (θ′ ) dθ′
La+1 (θ′ ) dθ′ϵa (θ) = ma eθ −−,′′R cosh(θ − θ ) 2π
R cosh(θ − θ ) 2π
23

TBA in the UV limit• constant solution ϵa (θ) → ϵ⋆a (θ → −∞)




πa
sin r+3
sin π(a+2)
r+3


Ya⋆ =
.
π
sin2 r+3• eﬀective central charge [Kirillov,Kuniba-Nakanishi-Suzuki]Zr
6 Xr(r + 1)
ceﬀ = 2.
maeθ La (θ)dθ =
π
r+3
R
a=1

• generalized paramermion SU (r + 1)2 /U (1)r [HISS]• TBA system for Homogeneous sine-Gordon model
SU (r + 1)2 /U (1)r [Fernandez-Pousa-Gallas-Hollowood-Miramontes]• PNP relation (quantum Matone relation)Pg(0) (1)(0) (1)ceﬀ
i=1 νi νDi − νi νDj = − 1224Wall-crossing and TBA
For a general polynomial potential, turning points are complex.
(V (q) = q 3 = E)complex periods (complex mass )ma = |ma |eiϕa ,a = 1, · · · , r.
e a (θ) = La (θ − iϕa ).
Leϵa (θ) = ϵa (θ − iϕa ),TBA system:e a−1 − Ka,a+1 ⋆ Le a+1e
ϵa = |ma |eθ − Ka,a−1 ⋆ L
1
1
2π cosh (θ + i(ϕs − ϕr ))Z(K ⋆ f )(θ) =K(θ − θ′ )f (θ′ )dθ′Kr,s =

R

This TBA formula is valid for ϕa satisfying |ϕa − ϕa±1 | < π2

25

When

π
,
2
the kernel picks the pole at θ′ = θ − iϕ2 + iϕ1 − πi2.
ϕ2 − ϕ1 >TBA equations are modiﬁed toe 2 − L2 θ − iϕ1 − iπ + iδ ,eϵ1 (θ) = |m1 |eθ − K1,2 ⋆ L
2


iπ
θ
e
e
ϵ2 (θ) = |m2 |e − K2,1 ⋆ L1 − L1 θ̂ − iϕ2 +− iδ .
2

Wall crossing phnenomena of the TBA equations[Gaiotto-Moore-Neitzke, AMSV, Toledo unpublished]26

3-term TBA
We can transform this 2-term TBA into 3-term TBA equations.
(n)Y-functions ϵ̃na (θ − iϕa ) = − log Ya (θ) (a = 1, 2, 12)cycles γ1 , γ2 and γ12 = γ1 + γ2
Y1 (θ)
Y2 (θ)
,,Y1n (θ) =Y2n (θ) =iπ
1 + Y2 θ − 2
1 + Y1 θ + iπ
2

iπ
Y
(θ)Y
θ
−
1
2
n
2
.
Y12
(θ) =
1 + Y1 (θ) + Y2 θ − iπ23-term TBA equations: [GMN, HISS]e2 − K + ⋆ L
e 12 ,eϵ1 (θ) = |m1 |eθ − K1,2 ⋆ L
1,12e 1 − K2,12 ⋆ L
e 12 ,eϵ2 (θ) = |m2 |eθ − K2,1 ⋆ L−
e 1 − K12,2 ⋆ Le2 .
e
ϵ12 (θ) = |m12 |eθ − K12,1⋆L

m12 = m1 − im2 = |m12 |eiϕ12 , f ± (θ) := f (θ ± i π2 )In general r-term TBA → · · · → r(r + 1)/2-term TBA

27

Example: cubic potential

cubic potential
cubic potential
κx2
V (x) =
− x3
2
m1 , m2 are represented by theelliptic integral.
TBA equations
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Z log 1 + e−ϵ2 (θ′ )dθ′cosh(θ − θ′ ) 2π
Z log 1 + e−ϵ1 (θ′ )
dθ′

,

R

ϵ2 (θ) = m2 eθ −Rcosh(θ − θ′ )2π.

• ceﬀ = 65 : SU (3)2 /U (1)228

PT-symmetric HamiltonianPT-symmetric Hamiltonian [Bender-Boettcher 1997]
q0<latexit sha1_base64="XAu/NLhDC/8JaZO0dMTpTEtJ/Kg=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSERRPBS8eK9oPaEPZbDft0s0m3Z0IJfQnePGgiFd/kTf/jds2B219YeHhnRl25g0SKQy67rdTWFvf2Nwqbpd2dvf2D8qHR00Tp5rxBotlrNsBNVwKxRsoUPJ2ojmNAslbweh2Vm89cW1ErB5xknA/ogMlQsEoWuth3HN75Ypbdeciq+DlUIFc9V75q9uPWRpxhUxSYzqem6CfUY2CST4tdVPDE8pGdMA7FhWNuPGz+apTcmadPgljbZ9CMnd/T2Q0MmYSBbYzojg0y7WZ+V+tk2J47WdCJSlyxRYfhakkGJPZ3aQvNGcoJxYo08LuStiQasrQplOyIXjLJ69C86LqWb6/rNRu8jiKcAKncA4eXEEN7qAODWAwgGd4hTdHOi/Ou/OxaC04+cwx/JHz+QP9DY2Q</latexit><latexit sha1_base64="fGXHp3f6fR1hTjxXeUm1P+AgPBw=">AAAB73icbZBNSwMxEIZn/az1q+rRS7AInspuERRPBS8eK9gPaJcym2bb0CS7JlmhlP4JLx4U8erf8ea/MW33oK0vBB7emSEzb5QKbqzvf3tr6xubW9uFneLu3v7BYenouGmSTFPWoIlIdDtCwwRXrGG5FaydaoYyEqwVjW5n9dYT04Yn6sGOUxZKHCgec4rWWe3uAKXEXrVXKvsVfy6yCkEOZchV75W+uv2EZpIpSwUa0wn81IYT1JZTwabFbmZYinSEA9ZxqFAyE07m+07JuXP6JE60e8qSuft7YoLSmLGMXKdEOzTLtZn5X62T2fg6nHCVZpYpuvgozgSxCZkdT/pcM2rF2AFSzd2uhA5RI7UuoqILIVg+eRWa1Urg+P6yXLvJ4yjAKZzBBQRwBTW4gzo0gIKAZ3iFN+/Re/HevY9F65qXz5zAH3mfP69Tj7I=</latexit>

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1Ĥ = p̂2 + iq̂ 3 − iλq̂.
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1

12

q+
<latexit sha1_base64="Hq7qOu4NKFlAyFWn5zMcEbCEgLc=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tFEISSiKB4KnjxWNF+QBvKZrtpl242cXcilNCf4MWDIl79Rd78N27bHLT1hYWHd2bYmTdIpDDout9OYWV1bX2juFna2t7Z3SvvHzRNnGrGGyyWsW4H1HApFG+gQMnbieY0CiRvBaObab31xLURsXrAccL9iA6UCAWjaK37x95Zr1xxq+5MZBm8HCqQq94rf3X7MUsjrpBJakzHcxP0M6pRMMknpW5qeELZiA54x6KiETd+Nlt1Qk6s0ydhrO1TSGbu74mMRsaMo8B2RhSHZrE2Nf+rdVIMr/xMqCRFrtj8ozCVBGMyvZv0heYM5dgCZVrYXQkbUk0Z2nRKNgRv8eRlaJ5XPct3F5XadR5HEY7gGE7Bg0uowS3UoQEMBvAMr/DmSOfFeXc+5q0FJ585hD9yPn8A9XmNiw==</latexit>• PT-symmetry: parity q → −q+ complex conjugation• three complex turning points• λ ≤ 0 real, positive discrete spectrum• λ > 0 and large, PT-symmetry is broken due tonon-perturbative eﬀect29

λ = 1 case1i
Πγ1 = Πp − Πnp , Πγ2 = Π∗γ1 , Πγ12 = Πγ1 + Πγ2223-term TBA
e2 − K + ⋆ L
e 12 ,eϵ1 (θ) = |m1 |eθ − K1,2 ⋆ L
1,12e 1 − K2,12 ⋆ L
e 12 ,eϵ2 (θ) = |m2 |eθ − K2,1 ⋆ L−
e 1 − K12,2 ⋆ Le2 .
e
ϵ12 (θ) = |m12 |eθ − K12,1⋆L

with ϕ1 = −α, ϕ2 = π2 + α, ϕ12 = 0.
quantum periods:1πs(Πγ1 )(ℏ) = −ie
ϵ1 (θ + i − iα),ℏ2

1
πs(Πγ1 )(ℏ) = −ie
ϵ2 (θ − i + iα)ℏ230BS vs exact quantization condition

Bohr-Sommerfeld quantization1s(Πp )(ℏ) = 2πℏ(n + ),2

n = 0, 1, 2, . . .

exact quantization condition


1
1
2 cos
s(Πp )(ℏ) + e− 2ℏ s(Πnp )(ℏ) = 0.
2ℏ
[Delabaere-Dillinger-Pham, Alvarez-Casares,Delabaere-Trinh]

31

Numerical calculation of s(Πp ) and s(Πnp ) and Voros spectrum

E=λ=1
15

Πp

Πnp

ℏ

ℏ

10

5

-5

-4

-3

-2

-1

1

horizontal axis θ = − log(ℏ)

nxn
En (x−1n )0 0. 560 405 699 850 0. 999 999 999 855
1 1. 496 238 960 118 0. 999 999 999 980
2 2. 509 053 231 083 0. 999 999 999 993
3 3. 507 747 393 173 0. 999 999 999 997
Voros sepctrum:xn = ℏn1(E) computed byusing EQC
En (x−1n ): computed by complex dilatation
technique32

ODE/IM and PT-symmetric Hamiltonian[Dorey-Dunning-Tateo, 0103051]
λ=02
H = p2 + iq 3πi
πiturning points: q0 = i, q+ = e− 6 , q− = −e 6 , α = π3
• Z3 -symmetry: eϵa (θ) = ϵ(θ)• periodicity ϵ(θ + 5π3 ) = ϵ(θ)
3-term=⇒ single term TBA [Dorey-Tateo 9812211]√

6πΓ(1/3) θϵ(θ) =e +3Γ(11/6)

Z

√
′

′

′

Φ(θ − θ )L(θ )dθ ,RΦ(θ) =

3 sinh 2θ
π sinh 3θ

TBA for Yang-Lee edge singularity (ceﬀ = 65 ) [Al Zamolodchikov 1990]
33

Numerical check
PT cubic oscillator with ℏ =
n
0
1
2

Ennum
1. 156 267 071 988
4. 109 228 752 810
7. 562 273 854 979

√

2

EnTBA
1. 156 267 071 988
4. 109 228 752 806
7. 562 273 854 971

Enp
1. 134 513 239 424
4. 109 367 351 095
7. 562 273 170 784

EnWKB
1. 094 269 500 533
4. 089 496 119 273
7. 548 980 437 586• Ennum : complex dilatation+Rayleigh-Ritz[Yaris-Bendler-Lovett-Bender-Fedders 1978]• EnTBA : TBA+exact quantization condition• Enp : TBA+Bohr-Sommerfeld quantization condition• EnWKB : Bohr-Sommerfeld approximation
34

Conclusions and outlook

Conclusions

We have provided an eﬃcient approach to solve the spectral
problem for arbitrary polynomial potentials in one-dimensional
Quantum Mechanics. The solution takes the form of a TBA
system for the (resummed) quantum periods which generalizes the
ODE/IM correspondence35Outlook• including angular momentum term ℓ(ℓ+1)
in the potential
x2[Dorey-Tateo, Bahzanov-Lukyanov-Zamolodchikov] ( Ito-Shu to appear)• ĝ∨ aﬃne Toda and ĝ BAE (Langlands dual) [Feigin-Frenkel,
Dorey-Dunning-Masoero-Suzuki-Tateo, Sun, Masoero-Raimondo-Valeri,
Ito-Locke](Ito-Kondo-Kuroda-Shu, work in progress)• massive ODE/IM [Lukyanov-Zamolodchikov,Dorey et al. , Ito-Locke,
Adamopoulou-Dunning, Negro,Ito-Shu]• application to Argyres-Douglas theory [Ito-Shu, Grassi-Marino,
Itoyama-Oota-Yano, Ito-Koizumi-Okubo, ...]

quantum periods and free energy, 2d/4d correspondence
Many new examples of ODEs
36