Neumann spectral determinants of Laplacians on hemispheres

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Numbers

$d$

convention

$D^N_d\text{ or }L^N_d$

$D^N_d$:

6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136

$L^N_d$:

1.837877066409345483560659472811235279722794947275566825634303080965531391854520795389486597271908395

$D^N_d$:

4.480706884083051472769819152578970124572257213683489429404394891300582270334913869667895592492946197

$L^N_d$:

1.499780820605574600208169056891178925389470234308186980150196153197485095260414743213793499300971709

$D^N_d$:

2.513461667539913506354081963449459501905534440035607856308615700410508465949105985811228846214245343

$L^N_d$:

0.9216609534035716323572688056613764911969342031577038786159657272057695915690797535261996568529187616

$D^N_d$:

1.936573341145783262660051050535362949974768164666039340131159035991269490008368375161709944162295889

$L^N_d$:

0.6609200922974785986905638705078894352275978350663596247999258078079614936260623979436308523594864327

$D^N_d$:

1.565600756194695541901049196815040252916793657812783022952515764630561827499099578997567080540090677

$L^N_d$:

0.4482696201059333821749616412913298104320565034779909504601175287564594626368031614697466576873696848

$D^N_d$:

1.330585599632418142217462745277892489593359743389149427029014851224239581001313882198020594621850867

$L^N_d$:

0.2856191458053538817496785928319235168639040955642695128591007120307714133391927932130975473929774609

$D^N_d$:

1.168808070707179737570142395406018907637705093264002254054690663591866023170440463774063399187816869

$L^N_d$:

0.1559844865530139122985450734944940493414223224126278158590265139770289780123182901451762919459115774

$D^N_d$:

1.038024294879871959028800170997614302889714238624900816054807344587121334687212136140056764163175848

$L^N_d$:

0.03731918994173141475049211856270437470001804521751160475198668653132792771818653070517258951139122351

$D^N_d$:

0.9454861089894831965807802409303630273500511466066309828809818113942771483435387647989371718457792757

$L^N_d$:

-0.05605608270018662508608190210713554626229121099945632606805837395811628191712029615694283841538016598

$D^N_d$:

0.8610170998324581971022796804730478909676576341233225239510134207751792490078194782189619344440207524

$L^N_d$:

-0.1496409143190341141302028915946166685129690119499237735793332267347396134092398683011239271893255390

$D^N_d$:

0.8003201795094489516749103302491918121476379507845093733575540852478772163117582881448528787946443277

$L^N_d$:

-0.2227434069958156223221248340691459374500018556549992061736162971772423612640886060657294100993623968

$D^N_d$:

0.7407549442146872793120730197699054876088353580226391505095690588811998351201692542361047528279751298

$L^N_d$:

-0.3000854179443422687317432558478682673565883851614361914138652359531147037075804230379346718016133379

$D^N_d$:

0.6975096160418753705619610784020695647356931200803930657675742684534916910160794219487574246147156712

$L^N_d$:

-0.3602389789416837591879247397296991100725096640509542625529846889553629645929461883366537797387335510

$D^N_d$:

0.6529922205380977750102441357836469489866235969465928071826458514503158846755781549662547736185249798

$L^N_d$:

-0.4261900631945920296270743277351528888516987924133600226009922456417969806841883559188572592423957820

$D^N_d$:

0.6204243818392869127662003210938526664035160355591595895198896164176731142777631937243225804003461672

$L^N_d$:

-0.4773515482597114203071572994730491847962139694487802065615972080725000632213785786974584053368317033

$D^N_d$:

0.5857482246793301612889676154943906707481437227313959813006492091570492353423971297669994628464259394

$L^N_d$:

-0.5348652324493070630152609146147400540200132446269819122195748870237953935370856526397759987654670109

$D^N_d$:

0.5602268136732349452395543206694874316793533357544540410297049191178047649443061020033831580587277240

$L^N_d$:

-0.5794135528366994833004613915803563663928877687283088299200141619627812735038005603384556529747608066

$D^N_d$:

0.5323667086592463292636083421249024256990431851210448543506219870189480461723981064896204559255721300

$L^N_d$:

-0.6304227251089309759072361558760725401161690735993201278224180871902525929267672237353957468077906447

$D^N_d$:

0.5117599058806279676302703349334373268367118096261278121524547630838762985804671239096910499505707056

$L^N_d$:

-0.6698996977533795491650628790638888964952669353297770662176045739126018837357913101925349400115140435

$D^N_d$:

0.4888306854461313875417235962198078067754710574652490360002434455904508269848541084366269404556685944

$L^N_d$:

-0.7157390959958643194374764702713373131249992068349076990056938372905055794075698278136553512163397101

Definition

The quantity $D^N_d$ is the zeta-regularized determinant of the unit round hemisphere's scalar Neumann Laplacian, omitting its constant zero mode [1]. The determinant $D^N_d=\det{}^{\prime}\Delta_N$ and its natural logarithm $L^N_d=\log D^N_d$ are one number in two invertible conventions.

Parameters

$d$

— dimension of the hemisphere ($d\geq1$)

convention

— determinant or natural logarithm

Formulas

(1)

$\displaystyle Z^N_d(s)=\sum_{k=1}^{\infty}\binom{d+k-1}{d-1}[k(k+d-1)]^{-s}$ for $\Re s>d/2$, where $k$ is the spherical-harmonic degree. [1], equations (1) and (2).

(2)

$D^N_d=e^{L^N_d}$ and $L^N_d=\log D^N_d=-(Z^N_d)^{\prime}(0)$.

(3)

$Z^N_1(s)=\zeta(2s)$, $D^N_1=2\pi$ and $L^N_1=\log(2\pi)$, where $\zeta$ is the Riemann zeta function.

(4)

$L^N_2=-\zeta^{\prime}(0)-2\zeta^{\prime}(-1)+\tfrac14$ and $L^N_3=\tfrac12\log(\pi/2)-\zeta^{\prime}(-1)-\zeta^{\prime}(-2)+\tfrac12$. These follow from the sphere and Dirichlet formulas of [2], Theorems 2.7, 2.8 and 3.2.

(5)

$Z^{S^d}(s)=Z^D_d(s)+Z^N_d(s)$, $D^{S^d}=D^D_dD^N_d$ and $L^{S^d}=L^D_d+L^N_d$. Here the superscript $S^d$ denotes the unit round sphere with its zero mode omitted, and $D$ denotes the Dirichlet hemisphere with the same radius and scalar operator. The quantities $Z$, $D$ and $L$ denote the spectral zeta function, determinant and natural log determinant, respectively. [1], equation (3).

Comments

(6)

The hemisphere is $H^d=\{x\in\mathbb{R}^{d+1}:|x|=1,\ x_{d+1}\geq0\}$ with its induced round metric. The nonnegative operator is $\Delta_N=-\operatorname{div}\operatorname{grad}$ on scalar functions, and the boundary condition is $\partial_\nu u=0$ on the equator, where $\nu$ is the outward unit normal. For $d=1$, $H^1$ is an interval of length $\pi$ with the Neumann condition at both endpoints. Radius is fixed to one, with no volume normalization or additive shift.

(7)

The spectral zeta function $Z^N_d(s)$ is continued meromorphically from $\Re s>d/2$ and is regular at $s=0$. Powers of positive eigenvalues use real logarithms. The prime on $\det{}^{\prime}$ omits the single constant zero mode; the prime on $Z^N_d$ differentiates with respect to $s$. [1], Section 2.

(8)

The positive determinant $D^N_d$ and its natural logarithm $L^N_d$ are one number in two invertible conventions, stored for the same dimensions.

(9)

Equatorial reflection separates the spherical harmonics into the Neumann and Dirichlet hemisphere sectors. Subtracting the Dirichlet log determinant from the sphere log determinant gives $L^N_d$. The finite expressions in [2], Theorems 2.7, 2.8 and 3.2, use Riemann zeta derivatives and elementary terms. [3].

Programs

(P1)

Sage

# Download generate.py and finite_formulas.py into the current directory.
from generate import logdet
for d in range(1, 21):
    L = logdet(d, working=150)
    print(d, L.exp(), L)

References

[1]

J. S. Dowker, Functional determinants on spheres and sectors, 1993, Section 2. (arXiv)

[2]

José Cunha and Pedro Freitas, Recurrence formulae for spectral determinants, 2024, Theorems 2.7, 2.8 and 3.2, and Appendix A. (arXiv)

Links

[3]

Cunha and Freitas: full text and finite determinant formulas

Similar tables

Dirichlet spectral determinants of Laplacians on hemispheres — uses the same hemispheres and scalar Laplacians with the Dirichlet boundary condition

Spectral determinants of Laplacians on spheres — holds the full-sphere determinants, each equal to the product of the Dirichlet and Neumann hemisphere determinants

Neumann spectral determinants of homogeneous oscillators — uses spectral zeta regularization for half-line Schrödinger operators with the Neumann boundary condition

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every integer dimension $1\leq d\leq20$ in both determinant and natural-logarithm conventions for the radius-one scalar Neumann problem with its constant zero mode omitted; this noncanonical practical prefix covers the interval, low-dimensional hemispheres and ten dimensions of each parity while keeping an independent spectral continuation feasible, rather than extending only the high-dimensional range)

How they were obtained:

The finite sphere and Dirichlet formulas of [2] were recomputed with exact combinatorial coefficients and Arb ball arithmetic, and their log determinants were subtracted using the spectral decomposition of [1]. No rounded database values enter the computation.

At 150 working decimal digits, all 40 values retain at least 504 bits of relative accuracy and overlap recomputations at 200 digits; 100 significant digits are stored. A separate direct continuation of the Neumann multiplicity-weighted spectral sum, with an exact geometric tail bound included in its Arb enclosure, agrees in every dimension and retains at least 560 bits of relative accuracy. Nonzero interval and low-dimensional controls, zeta functional-equation checks, changed tail split points, and deliberate perturbations test the comparisons. Every stored decimal interval is checked against the independent enclosure, and the determinant product and logarithm sum identities are checked using values read from the database. The direct continuation resolves the conflicting three-dimensional expression on printed page 7 of [1]; the calculation is preserved in the attached derivation. The generator, selection rationale and numerical evidence are attached.