Parity-twisted spectral zeta values of homogeneous oscillators

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Numbers

$N$

$n$

$Z_N^{\rm P}(n)$

0.8402639322374488944048153024634379436012556119531050340213474817187877358778344659048994583215183971

0.9159655941772190150546035149323841107741493742816721342664981196217630197762547694793565129261151062

0.7836009674832748802994658836068563594913551043417053863467884304524923110139765593281934206754694301

0.8890409906909802662720814146754651243609311125757582310392653924167659983027543539583655055347313320

0.7633029347734727834959000527236241533633885665433324749526156260786368657836555348694155656464276850

0.8331557979080023912186327602786739818027795159719601522088453449873976174341655496580463554931827076

0.7381576086874983492564476162301493852595983745750761414712027446306248869928332434227605627087190228

0.7737981993082750289749985800694865178514515738750053921462175838371366449864997501693924133393719765

0.7130049393708506363245512906282431946383397889032182210844879319674173567477870386640916783824397821

0.7189522955983526636449399098905563006054028970317956482942253986417211255175249609696733568525455117

0.6895095879155625995697201942669598753289758766705897224101520104140409660322785363579529907389043769

0.6705487969197929633016906018890316494457610489096220304531396549097146058244081547431706847838743433

0.6681353096862297134082697452146855807561545374458468758781213248747711875833619150973618514565101788

0.6284843151105194727871486205751724570877879738355925876550839105783826271355334719450040249322967084

0.6488860383801290230087502862946291231713276127786702195800192069501654980792955401605737150650559535

0.5920448685693137925245104177161908634105999951384435978396261361641418377933074286803394712316773001

0.6315987101847940568232165147790744252766563060852571996248286598836618533276543221855544061835901085

0.5604096453376910409790981611657337469997907721761708043408046267212269110837353463527614469024849745

0.6160621130412225937872363358946895448717726769629583114819343483919118150321189743537096505037674974

0.5328199648061817691759353378159151592624568848282052319811243189639021335559765829244720414309756464

0.6020647751346410782612229259678147963509492093401109792468081395793212982997435527282514297178344161

0.5086249291621045125211097258333361978071782366909665844455067587461397968775632024814745273269731364

Definition

The parity-twisted spectral zeta values are $Z_N^{\rm P}(n)=\sum_{k=0}^{\infty}(-1)^k E_{N,k}^{-n}$, where $E_{N,0}<E_{N,1}<\cdots$ are the eigenvalues of $H_N=-d^2/dx^2+|x|^N$ on $L^2(\mathbb R)$, with the even ground state indexed by $k=0$. [1]

Parameters

$N$

— degree of the potential ($N\geq1$)

$n$

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

Formulas

(1)

For $N>2$ and $\nu=1/(N+2)$, $\displaystyle Z_N^{\rm P}(1)=\frac{\sin(\pi\nu)}{2\sqrt{\pi}}(2\nu)^{2-4\nu}\frac{\Gamma(\nu)\Gamma(2\nu)\Gamma(3\nu)}{\Gamma(1/2+2\nu)}$, where $\Gamma$ is the Gamma function. [3], equation (C.22).

(9)

$Z_1^{\rm P}(2)=\kappa^{-1}-\kappa^2$, where $\kappa=-\operatorname{Ai}^{\prime}(0)/\operatorname{Ai}(0)=3^{1/3}\Gamma(2/3)/\Gamma(1/3)$. Here $\kappa^{-1}$ is the Neumann second moment and $\kappa^2$ is the Dirichlet second moment. [4]

(3)

$E_{2,k}=2k+1$ and $Z_2^{\rm P}(n)=\sum_{k=0}^{\infty}(-1)^k(2k+1)^{-n}$ for integers $n\geq2$. In particular, $Z_2^{\rm P}(2)=G$, where $G$ is Catalan's constant. [1], Section 2.1.

(4)

$\displaystyle Z_6^{\rm P}(2)=\frac{(\sqrt{2}-1)\pi^5}{32\Gamma(3/4)^4\Gamma(7/8)^2\Gamma(5/8)^2}$. [2], Example 2.2.

Comments

(5)

The kinetic coefficient and potential coefficient are both $1$, and the additive energy shift is $0$. The even-indexed levels form the half-line Neumann spectrum with $y^{\prime}(0)=0$; the odd-indexed levels form the half-line Dirichlet spectrum with $y(0)=0$. Thus $Z_N^{\rm P}(n)=Z_N^+(n)-Z_N^-(n)$. [1], equations (2), (3), and (5).

(6)

The asymptotic growth $E_{N,k}\asymp k^{2N/(N+2)}$ gives absolute convergence for $n>(N+2)/(2N)$. The entries use this domain, in which both boundary traces converge separately. No analytic continuation, finite-part value, or conditionally convergent first moment is included. [1]

(7)

The full spectral zeta values add the two boundary traces, $Z_N(n)=Z_N^+(n)+Z_N^-(n)$. The associated zeta-regularized determinants are in the Neumann determinant table and the Dirichlet determinant table.

(8)

These integer orders are traces of inverse powers of $H_N$ with the parity operator inserted. They also give the coefficients of the logarithm of the ratio of the Neumann and Dirichlet spectral determinants, starting at the first absolutely convergent order. [1], equation (6).

(9)

At $N=1$, the Neumann eigenvalues are the absolute values of the negative zeros of $\operatorname{Ai}^{\prime}$ and the Dirichlet eigenvalues are the absolute values of the negative zeros of $\operatorname{Ai}$. Their inverse-power sums are in the Airy derivative zeta table and the Airy zeta table.

(10)

For $N>2$, the boundary contributions to the second moment have generalized hypergeometric formulas. [2], equation (2.5b); [3], equations (C.28) and (C.33). [5]

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)
P = mp.sin(mp.pi*v)/(2*mp.sqrt(mp.pi))*(2*v)**(2-4*v)
P *= mp.gamma(v)*mp.gamma(2*v)*mp.gamma(3*v)/mp.gamma(mp.mpf(1)/2+2*v)
print(P)

References

[1]

A. Voros, Exact sum rules for spectral zeta functions of homogeneous 1D quantum oscillators, revisited, Journal of Physics A: Mathematical and Theoretical 56 (2023), 064001, Sections 1.1, 2, and 3. (arXiv) (doi)

[2]

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

[3]

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. (zbMATH) (MR)

Links

[4]

NIST DLMF: Airy differential equation and initial values

[5]

Numdam: The return of the quartic oscillator, Appendix C

Similar tables

Full spectral zeta values of homogeneous oscillators — adds the two parity sectors instead of subtracting them

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 with the same operator scale

Neumann spectral determinants of homogeneous oscillators — uses the even-parity spectrum with the same operator scale

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

Data properties

Entries are of type: real number

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 $3\leq N\leq12$; this noncanonical practical degree prefix covers the linear and harmonic controls and several steeper wells of both parities, while the first two inverse-operator moments give breadth across degrees with explicit Gamma and hypergeometric formulas; the omitted first moments are not absolutely convergent)

How they were obtained:

Every entry was computed at 150 and 200 decimal working digits. For degrees greater than two, the boundary traces use Watkins equations (2.5a,b), with the factor 1/4 in (2.5a). Second moments use a Gamma-ratio asymptotic tail with cutoff and correction count (160,140) and (320,180). The minimum agreement was over 128 significant digits, and 100 digits are stored. The asymptotic remainder is not enclosed, so this is heuristic (agreement-checked) rigour.

All 22 values were compared with fresh evaluations of different formulas: Voros equations (C.22), (C.28), and (C.33), Airy logarithmic derivatives, and harmonic Hurwitz zeta values. The minimum agreement exceeded 157 digits. That high-precision comparison shares the generic tail helper; a further evaluation with mpmath’s own hypergeometric routine at 25 working digits agreed to over 24 digits and shares no tail code. The first two quantum-Wronskian identities, the sextic closed form, the cubic Neumann second-moment closed form, a known nonzero hypergeometric control, and a deliberate perturbation were checked. The attached generator, checks, argument-selection note, and numerical evidence state the methods and their limitations.

After filling, verify() matched every entry. Fresh alternate evaluations lie inside all 22 stored rounding intervals. The first two quantum-Wronskian identities were repeated using the stored parity traces and the stored full traces in T541, with rounding intervals propagated through the identities.