Neumann spectral determinants of homogeneous oscillators

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Numbers

$N$

$D_N^+(0)$

0.9174908978832602608991682695065397930724593210731346783920654448634928647913427574433215477044716666

comment: Linear oscillator with a Neumann boundary.

0.9777410674469237976315354682247592341449847144727670865937397361905038778202402753109639597981096774

comment: Harmonic oscillator with a Neumann boundary.

1.064522252385131096750040731747761746215912566486674920772736137745562416850056549331466072350005362

1.157233039336957033847151690474598542143789519361469669724941687697572350186100631584842928602635581

comment: Pure quartic oscillator with a Neumann boundary.

1.249998366723245384586811145684451967054222495904290335592706234592151300135174888033718106720073989

1.340823989385706702311248976750646073733490627573834655269072143078667932240025123173550458955543983

1.429028741995882917643318273432572421327213730693785834602188602822035276639772242922417610395275229

1.514439394742978509113412404411454292596258246497996964053907676904897471625664696923870313472825033

1.597092082453000705953602769321551907080706605559096914913091327716249799110748955408598686877674922

10:

1.677109090164437320824534956274348246850551361475598785449165457413528851530569439085690250732509739

11:

1.754644480817827754843085600704559310942525507866086034997564835797028990224413670087341705681183759

12:

1.829859253270710126866219295980763637262605438431146646912608291895091186894045073479769636448801414

Definition

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

Parameters

$N$

— degree of the potential ($N\geq1$)

Formulas

(1)

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

(2)

$D_N^+(0)D_N^-(0)=1/\sin(\pi/(N+2))$, where $D_N^-(0)$ is the Dirichlet determinant at the same degree and normalization. [1], equation (136).

(3)

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

(4)

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

Comments

(5)

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 $s=0$, where it is regular. The determinant is $D_N^+(0)=\exp[-(Z_N^+)^{\prime}(0)]$.

(6)

The kinetic coefficient and the coefficient of $x^N$ are both $1$. The argument $0$ is the fixed additive energy shift, and the prime on $Z_N^+$ differentiates with respect to $s$. Powers of positive eigenvalues use their real logarithms.

(7)

This is the even-parity spectrum of $-d^2/dx^2+|x|^N$ on the full line. The Dirichlet spectral determinants use its odd-parity spectrum. In the pure anharmonic-oscillator eigenvalue table, the full-line level index starts at zero, so its even indices select the Neumann spectrum for $N=4,6,8,10$.

(8)

For $N=1$, the eigenvalues are the absolute values of the negative zeros of $\operatorname{Ai}^{\prime}$, the arguments of the Airy extrema. Their convergent inverse-power traces are the Airy derivative zeta values.

(9)

For $N=2$, the Neumann eigenvalues are $E_{2,j}^+=4j+1$, 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(1-nu).gamma() * B(nu)**(nu-QQ(1)/2) / 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

Dirichlet spectral determinants of homogeneous oscillators — uses the same potentials and scale with the Dirichlet boundary condition

Eigenvalues of the pure quartic, sextic, octic and decic oscillators — gives individual full-line eigenvalues for four even degrees; even-indexed levels form the Neumann spectra

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

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 practical prefix of the homogeneous-oscillator family that includes the linear and harmonic controls, cubic and quartic wells, and several steeper wells of both parities; this bounded low-degree reference gives breadth across the family without a long listing of high degrees)

How they were obtained:

Each determinant was evaluated in Arb ball arithmetic with exact rational inputs, at 429 working bits for 100 significant digits. The lowest measured relative accuracy was 425 bits.

All twelve values agree with 200-digit recomputations and with an independent 720-bit reconstruction of the zero-energy boundary derivative using modified Bessel functions and an exact rational power series with explicit tail bounds. Those independent enclosures retain at least 708 bits. A nonzero elementary control tests the reconstruction. Airy and harmonic-oscillator evaluations test the boundary condition and scale, and controlled numerical differentiation of the harmonic spectral zeta function supplies a further heuristic check. The product identity with the Dirichlet determinant holds for every degree. Deliberate perturbations of size 10^(-90) are rejected. The attached checks test the stored rounding intervals against independent enclosures and repeat the product identity using both stored tables.