Dirichlet spectral determinants of homogeneous oscillators

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Numbers

$N$

$D_N^-(0)$

1.258541682585905453517532273632992728014593496699297507604505531690623812368982320965612763719882503

comment: Linear oscillator with a Dirichlet boundary.

1.446409084632077142535701449843129003398744062829433434874645922986286342445006721388574729409513177

comment: Harmonic oscillator with a Dirichlet boundary.

1.598183234678470396487150143591480967260802571121156156204932012070621528536593973547218456896655294

1.728260369359926745251607253272653555129581919787799560671977046811593157483019537464135895813326940

comment: Pure quartic oscillator with a Dirichlet boundary.

1.843814305937225775891067926461126001620261247452675289767666435840204100779995363158244994885885837

1.948895567530788691170989935306182552691814208952929456179612422484739021198322125920832025162009817

2.046008113230455119941431652737474478849006452836844549320499006301642466616521691194461770682158779

2.136809164323802850865353261070144805618869940529671486273943587674099204413629085542314608068853067

2.222455155768187930444232571206697866496308044072120809620423355499661685163374496995571353654119721

10:

2.303787706962726240257587095809052735002637655336595288601362011585631393134439510828085865508758411

11:

2.381440522306130547687173784846882388868929042584832744362881435532131923959456520087285171885206821

12:

2.455904299416680850936264027035181026788158242481472750497332397571314660224454288807729941395996598

Definition

The zeta-regularized determinant $D_N^-(0)$ of $H_N^-=-d^2/dx^2+x^N$ on $x\geq0$, with the Dirichlet boundary condition $y(0)=0$ and square-integrability at infinity [1].

Parameters

$N$

— degree of the potential ($N\geq1$)

Formulas

(1)

With $\nu=1/(N+2)$, $\displaystyle D_N^-(0)=\frac{\Gamma(\nu)\nu^{1/2-\nu}}{\sqrt{\pi}}$, where $\Gamma$ is the Gamma function. [1], equation (136).

(2)

$D_1^-(0)=2\sqrt{\pi}\operatorname{Ai}(0)$, where $\operatorname{Ai}(0)$ is the initial value of the Airy function. [1], equation (148).

(3)

$Z_2^-(s)=4^{-s}\zeta(s,3/4)$ and $D_2^-(0)=\sqrt{\pi}/\Gamma(3/4)$, where $\zeta(s,a)$ is the Hurwitz zeta function. [1], equation (160); [2].

Comments

(4)

Let $E_{N,j}^-$ be the positive eigenvalues of $H_N^-$ in increasing order, indexed from $j=0$ and counted once. The spectral zeta function is $Z_N^-(s)=\sum_{j=0}^{\infty}(E_{N,j}^-)^{-s}$ for $\Re s>(N+2)/(2N)$, continued meromorphically to $s=0$. The determinant is $D_N^-(0)=\exp[-(Z_N^-)^{\prime}(0)]$.

(5)

The kinetic coefficient and the coefficient of $x^N$ are both $1$. The argument $0$ of $D_N^-$ is the fixed additive energy shift, and the prime on $Z_N^-$ differentiates with respect to $s$. The continued spectral zeta function is regular at $s=0$.

(6)

The same spectrum is the odd-parity sector of $-d^2/dx^2+|x|^N$ on the full line. The corresponding full-line eigenvalues for $N=4,6,8,10$ are in the pure anharmonic-oscillator table; its odd energy-level indices select the Dirichlet sector.

(7)

For $N=1$, the eigenvalues are the absolute values of the negative Airy zeros. The convergent inverse-power traces of this spectrum are the Airy zeta values.

(8)

For $N=2$, the half-line Dirichlet eigenvalues are $E_{2,j}^-=4j+3$, with $j=0,1,\ldots$. The full-line harmonic oscillator has eigenvalues $2k+1$, with $k=0,1,\ldots$.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

B = RealBallField(429)
for N in range(1, 13):
    nu = QQ(1) / (N + 2)
    print(N, B(nu).gamma() * B(nu)**(QQ(1)/2-nu) / B.pi().sqrt())

References

[1]

A. Voros, Exercises in exact quantization, Journal of Physics A: Mathematical and General 33 (2000), 7423–7450, Appendix A.1 and A.2.2–A.2.3. The corrected preprint is version 2. (arXiv) (doi)

Links

[2]

NIST DLMF 25.11: Hurwitz zeta values and derivatives at zero, equations (13) and (18)

Similar tables

Eigenvalues of the pure quartic, sextic, octic and decic oscillators $-y''+x^{2m}y$ — gives individual full-line eigenvalues for four even degrees; odd-indexed levels are the Dirichlet spectra used here

Airy zeta values — gives convergent inverse-power traces of the degree-one Dirichlet spectrum

Values of the Airy function of the first kind $\operatorname{Ai}(x)$ — its initial value determines the degree-one determinant

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it holds every integer degree $1\leq N\leq12$, a noncanonical reference prefix of the homogeneous-oscillator family that includes the linear and harmonic controls and several steeper wells of both parities; this practical cutoff covers low-degree spectral models without extending into a long listing of high degrees)

How they were obtained:

Each determinant was evaluated with Arb using exact rational arguments in the Gamma formula, at 429 working bits for 100 significant digits. The lowest measured relative accuracy was 424 bits.

All twelve values agree with 200-digit recomputations and with an independent 620-bit reconstruction of the zero-energy boundary value from modified Bessel functions and an exact rational Frobenius series with explicit tail bounds. The independent enclosures retain at least 550 bits. A nonzero elementary control tests that reconstruction; Airy and harmonic-oscillator evaluations test the boundary condition and spectral scale. The harmonic spectral-zeta derivative was also evaluated at 150 and 200 decimal digits with mpmath. Deliberate perturbations of size 10^(-90) are rejected. Every stored rounding interval is checked to contain the independent enclosure. The attached generator and checks reproduce these computations.