Full spectral zeta values of homogeneous oscillators

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Numbers

$N$

$n$

$Z_N(n)$

1.903178396159447800139065168030632969360008134672767094447247485590632327183550118473575762270501069

1.233700550136169827354311374984518891914212425905098828301668672027505602802400655375221675464819029

3.319386965493754693169476319946829002023560536117475592219264782156101837521626859818290822847120283

1.098003370680573465880414073379908567918383166689158210595599027328569244936287063325952316107760017

2.289908804320418350487700158170872460090165699629997424857846878235910597350966604608246696939283055

0.9963208276787715060724700465937150050135078031028254076131607718349949712416468399359828502501163302

1.922070491443430806944571952871298785373765682468328621508611869298918480339198212887921576327099698

0.9108397549065565510669343408855582684034704619311029148439157228421820402300559971170081979305731005

1.721346194668113966341474187559077603311862291076695736298545207880918722674511398540029647836713598

0.8387491545184126620672059374086251999303201267569848904935731531423690903595456176220196457332217374

1.589600430045033617309990840297260220894966166112886046442566142856265279624410293902798433481082535

0.7779567496716873852938031451573028143153605950969387636150390100708468296456187824033540073685081913

1.493995970626283323606449483976215692057478061489339631271175745340327534965228986029209139976675529

0.7264905365509530523276104496548539836689873482459224335917337074144197417298946322935003813605340313

1.420218357421152822820077921486188836704305652818475236813235253633683930709727361091504138335855076

0.6826375530927969955090308776399444955919809295822688294987029186446595921535004248801228803211452467

1.360906080874816610219754929814874604665365898681591710881641854039578548249194519520555462377414348

0.6449883867695178073696874156734389647644144971529971375192277476286622972307630513117654251004663053

1.311818983165571593593679839008201567953562395209460172667092562748054778746015293732248740728401489

0.6124120914440976069306457015509485417142925844630748902282096534948009200846493142461269316146061478

1.270306261143288234259549964910521536697501322389553580887760188612699158305650179350427774969227126

0.5840093424641228311308246685518228896715669841124739760982547285404538520450949108939539694109215394

Definition

The full spectral zeta values are $Z_N(n)=\sum_{k=0}^{\infty}E_{N,k}^{-n}$, where $E_{N,0}<E_{N,1}<\cdots$ are the eigenvalues of $H_N=-d^2/dx^2+|x|^N$ on $L^2(\mathbb R)$, each counted once. [1]

Parameters

$N$

— degree of the potential ($N\in\mathbb Z_{\geq1}$)

$n$

— inverse-power order ($n\in\mathbb Z$, $n>(N+2)/(2N)$)

Formulas

(1)

With $\nu=1/(N+2)$ and $N>2$, $\displaystyle Z_N(1)=\frac{\tan(2\pi\nu)}{\tan(\pi\nu)}\frac{\sin(\pi\nu)}{2\sqrt{\pi}}(2\nu)^{2-4\nu}\frac{\Gamma(\nu)\Gamma(2\nu)\Gamma(3\nu)}{\Gamma(2\nu+1/2)}$, where $\Gamma$ is the Gamma function. [3], equation (140).

(8)

$Z_1(2)=\kappa^2+\kappa^{-1}$, where $\kappa=-\operatorname{Ai}^{\prime}(0)/\operatorname{Ai}(0)=3^{1/3}\Gamma(2/3)/\Gamma(1/3)$. The two terms are respectively the Dirichlet and Neumann second moments. [4]

(3)

$E_{2,k}=2k+1$ and $Z_2(n)=(1-2^{-n})\zeta(n)$ for integers $n\geq2$, so $Z_2(2)=\pi^2/8$. Here $\zeta$ is the Riemann zeta function. [3], Appendix A.2.3.

Comments

(4)

The kinetic coefficient and potential coefficient are both $1$, the additive energy shift is $0$, and the ground-state index is $k=0$. The potential is $|x|^N$ for odd as well as even degrees, and all eigenvalues are positive.

(5)

The even-indexed eigenvalues form the half-line Neumann spectrum, and the odd-indexed eigenvalues form the half-line Dirichlet spectrum. Their inverse-power sums are added here, with no parity sign. The associated regularized determinants are in the Neumann determinant table and the Dirichlet determinant table. [1]

(6)

The asymptotic growth $E_{N,k}\asymp k^{2N/(N+2)}$ gives absolute convergence for $n>(N+2)/(2N)$. No analytic continuation or finite-part value is used in these entries. [1]

(7)

Integer orders are traces of inverse powers of the operator. They also give the coefficients of the expansion of the logarithmic spectral determinant at zero energy, starting at the first convergent order. [1], equation (2.9); [3], equations (9)–(11).

(8)

For $N=1$, the two parity sectors give the Airy zeta values and the Airy derivative zeta values. The full trace is their sum.

Programs

(P1)

Python

# First moment for any integer degree N > 2.
from mpmath import mp
mp.dps = 120
N = 4
v = mp.mpf(1)/(N+2)
A = mp.sin(mp.pi*v)/(2*mp.sqrt(mp.pi))*(2*v)**(2-4*v)
A *= mp.gamma(v)*mp.gamma(2*v)*mp.gamma(3*v)/mp.gamma(2*v+mp.mpf(1)/2)
print(A*mp.tan(2*mp.pi*v)/mp.tan(mp.pi*v))

References

[1]

J. Watkins, Spectral zeta functions of a 1D Schrödinger problem, Journal of Nonlinear Mathematical Physics 19 (2012), 1250026, Section 2. (arXiv) (doi)

[2]

A. Voros, The return of the quartic oscillator. The complex WKB method, Annales de l’Institut Henri Poincaré, Physique théorique 39 (1983), 211–338, Appendix C, especially equations (C.22), (C.23), and (C.28). (zbMATH) (MR)

[3]

A. Voros, Exercises in exact quantization, Journal of Physics A: Mathematical and General 33 (2000), 7423–7450, Appendix A. The corrected preprint is version 2. (arXiv) (doi)

Links

[4]

NIST DLMF: Airy differential equation and initial values

[5]

Numdam: The return of the quartic oscillator, Appendix C

Similar tables

Eigenvalues of pure anharmonic oscillators — gives individual full-line levels for four even degrees, while each trace here sums the entire infinite spectrum

Airy zeta values — gives the Dirichlet contribution at degree one

Airy derivative zeta values — gives the Neumann contribution at degree one

Dirichlet spectral determinants of homogeneous oscillators — uses the odd-parity spectrum of the same potentials

Neumann spectral determinants of homogeneous oscillators — uses the even-parity spectrum of the same potentials

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds order $n=2$ for every degree $1\leq N\leq12$ and order $n=1$ for every degree $3\leq N\leq12$; the degree cutoff is a noncanonical practical prefix covering linear and harmonic controls and several steeper wells of each parity, while the first two inverse-operator moments give breadth across the degrees with explicit Gamma and hypergeometric formulas; the omitted first moments diverge)

How they were obtained:

Every value was computed at 150 and 200 decimal working digits. The first moments use the Gamma formula in Watkins equation (2.5a), including its factor 1/4. The second moments use equation (2.5b), with a Gamma-ratio asymptotic tail, at two cutoffs and two correction counts. The minimum agreement was 129 significant digits, and 100 digits are stored. The tail is not certified, so the rigour is heuristic (agreement-checked).

All entries were compared with separate full-trace formulas from [2] and [3], with a third cutoff and working precision, agreeing to at least 156 digits. That check shares the generic summation helper. A further comparison using mpmath’s own hypergeometric implementation agreed to at least 24 digits. Airy log derivatives and the harmonic odd-integer spectrum supply independent controls for degrees one and two. A known nonzero hypergeometric control, the first two quantum-Wronskian identities for every degree from 3 through 12, and a deliberate numerical perturbation were tested. The attached generator, checks, argument-selection note and numerical evidence record the methods, tolerances and limitations.