Dirichlet spectral determinants of Laplacians on hemispheres

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Numbers

$d$

convention

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

$D^D_d$:

6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136

$L^D_d$:

1.837877066409345483560659472811235279722794947275566825634303080965531391854520795389486597271908395

$D^D_d$:

0.7131266491477017280095075738643492893842210989264195585953388211606338613672803718727323049800589736

$L^D_d$:

-0.3380962458037708833524904159200563543333247129673798454841069277680462965941060521756930979709366865

$D^D_d$:

1.328387560976645518420403226535296241318700408799215111873508642392151547768179093015233578674891213

$L^D_d$:

0.2839658465626150833466616066339917737443615768475555654036011239844438142056201236130127831843199381

$D^D_d$:

0.8969159320955258953479727366837040804197412575296640670030107481115890343357463350130033497763114209

$L^D_d$:

-0.1087931425021504642551018958519749164947070379314316853526548293791134229547863292500345749795643697

$D^D_d$:

1.126033786829180886933293203570488268482233083749086373094307015164288032235742468152490092632934312

$L^D_d$:

0.1187015353323175831620816769431807596357004131057489764996003210629248640202732046474685339735057396

$D^D_d$:

0.9695117445909840188184863871660625985448083245540513218732937771499084630869765084827637092358063743

$L^D_d$:

-0.03096269030268254829589712139409311322827110000209533189891376123993889812357963719377738085752802163

$D^D_d$:

1.045955511755983387646053150027294391477784481614067014413011628051599162336370352740429201689247421

$L^D_d$:

0.04493083295618661906863018032877043296895060091160968595120192777766734696978946841549438939890232637

$D^D_d$:

1.011931145867773764960480344677118727421387052435799755325254266079591628148791165865459305096609509

$L^D_d$:

0.01186053087054040730780428371471654373405229843142950563931470252190765834626782750121605648034740566

$D^D_d$:

1.001319229530130242416134784527835962310700099015345805006660252987205729842965022904625281656448451

$L^D_d$:

0.001318360111411554660014809299306140912597109181213285160774723109737233820126192693949347806096857014

$D^D_d$:

1.040842116541575302637906607985979480045324456183305178582191822237253010927516159976917447408042752

$L^D_d$:

0.04003011294469789666858859499221844592856486952735251218291613370762078763798751091334044508837810791

$D^D_d$:

0.9721713693813062185727874733456223256423395628067151718508990224688889170328963773558451370013529737

$L^D_d$:

-0.02822318411391918460849182736883561800248909405725536404529832756858459478377538027347952287800775922

$D^D_d$:

1.062299914041009576701292167761337839783101982199139683164840384203668904540396833150971934756819655

$L^D_d$:

0.06043628788601660482380991786117140101533863558123100392792060356588314474414728519261816730899334755

$D^D_d$:

0.9513068006443022420486424437001151969964560845713002672326859218787839047458434921674338744137464993

$L^D_d$:

-0.04991866000488374779838327070185415635297989078668165888127506962319186349096919715705414171674629633

$D^D_d$:

1.079116795783014659122007421521716481036990155190190447396044404808196680197263151823317035335033603

$L^D_d$:

0.07614292488788080780645421770821934901825362383983178810853687326245614444622045609406761046886907455

$D^D_d$:

0.9354485469154419990097193491544669508138734856417803830013119973720875690568860298241911101587695888

$L^D_d$:

-0.06672913540544414828543420131375776820786731962093851077448313531941522341977680148615079423758266163

$D^D_d$:

1.092804075027248285545614289863744622796701201551728383400677811887405580756488531169115056673142035

$L^D_d$:

0.08874693880551819600741622447092178766338531366927405927225047009744238625711193779663432499447205199

$D^D_d$:

0.9228760953414427335297702634770443079002175662266422839061170162879286634166485355014728342475720075

$L^D_d$:

-0.08026029472398557473527280572973865418864278619338719503569327340052886212559772527930930436843645871

$D^D_d$:

1.104257764534915908727057329007965720355436952826834441874701714188694266495884946789472316804102575

$L^D_d$:

0.09917340297072054327371607050328974759430468022407745761415644247877039945690909915267041382639096327

$D^D_d$:

0.9125923120214811315904064572894285520072129934797038124664050738682117261911038450551952633864354813

$L^D_d$:

-0.09146603478305119677951394112579355237439855183416979791074764959704892823747937098632525838960777588

$D^D_d$:

1.114048199076945149312817857989661120012285298229990382480466541159334002897694082824534066036645308

$L^D_d$:

0.1080004072450099672224738846965950463725837865792066218531359390631396693912207623684558012113636165

Definition

$D^D_d$ is the zeta-regularized scalar Dirichlet Laplacian determinant on the unit round hemisphere $H^d$ [1]. The positive determinant $D^D_d$ and its natural logarithm $L^D_d$ are one number in two invertible conventions.

Parameters

$d$

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

convention

— determinant or natural logarithm

Formulas

(1)

$\displaystyle Z^D_d(s)=\sum_{k=1}^{\infty}\binom{d+k-2}{d-1}[k(k+d-1)]^{-s}$ for $\Re s>d/2$, where $k$ is the spherical-harmonic degree. [1], Section 3.1.

(2)

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

(3)

$Z^D_1(s)=\zeta(2s)$, $D^D_1=2\pi$, and $L^D_1=\log(2\pi)$, where $\zeta$ denotes the Riemann zeta function.

(4)

$L^D_2=\zeta^{\prime}(0)-2\zeta^{\prime}(-1)+1/4$ and $L^D_3=-\zeta^{\prime}(0)+\zeta^{\prime}(-1)-\zeta^{\prime}(-2)-1/2$, with Riemann zeta derivatives. [1], Theorem 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 $Z^{S^d}$, $D^{S^d}$ and $L^{S^d}$ refer to the unit round sphere with its zero mode omitted. The Neumann hemisphere quantities $Z^N_d$, $D^N_d$ and $L^N_d$ use eigenvalues $k(k+d-1)$ with multiplicities $\binom{d+k-1}{d-1}$ for $k\geq1$ and omit the constant zero mode. Equatorial reflection splits the sphere spectrum into these two boundary conditions. [2], 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 operator is the nonnegative scalar Laplace–Beltrami operator $-\operatorname{div}\operatorname{grad}$, with Dirichlet condition $u=0$ at the equator. There is no volume normalization, curvature shift or mass term. For $d=1$, the hemisphere is an interval of length $\pi$, and the condition holds at both endpoints.

(7)

The spectral zeta function $Z^D_d(s)$ is continued meromorphically from $\Re s>d/2$ and is regular at zero. Every eigenvalue is positive, so no zero mode is removed and all logarithms are real. The prime on $Z^D_d$ denotes differentiation with respect to $s$.

(8)

The determinant $D^D_d>0$ and its natural logarithm $L^D_d$ determine one another by exponentiation and logarithm. Both conventions use the same dimensions.

(9)

The finite formulas in [1], Theorem 3.2, express $L^D_d$ through Riemann zeta derivatives at nonpositive integers and rational terms. The full text and rounded comparison values are available in [3].

(10)

The unnumbered $d=3$ Neumann-minus-Dirichlet zeta-derivative difference on page 7 of [2] has a constant term $+1/2$. Substitution in its Barnes formula (8), together with the sphere formula, gives $-1$, consistent with [1], Theorem 3.2. The attachment source-check-notes.md gives the substitution.

Programs

(P1)

Sage

# Download the attached generate.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]

José Cunha and Pedro Freitas, Recurrence formulae for spectral determinants (2024), Section 3.1, Theorem 3.2. (arXiv)

[2]

J. S. Dowker, Functional determinants on spheres and sectors (1993), Section 2, equations (1)–(8). (arXiv)

Links

[3]

Cunha and Freitas, Section 3.1 and Appendix B, Table 2

Similar tables

Spectral determinants of Laplacians on spheres — holds the determinants on the full unit round spheres; their spectra split into Dirichlet and Neumann hemisphere sectors

Dirichlet spectral determinants of homogeneous oscillators — uses the same spectral zeta regularization for half-line Schrödinger operators with a Dirichlet 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 Dirichlet hemisphere, which has no zero mode to remove; this noncanonical practical prefix includes the interval and ten dimensions of each parity, while keeping multiplicity degree at most 19 for an independent spectral continuation instead of extending only high-dimensional depth)

How they were obtained:

The finite formulas of [1], Theorem 3.2, were evaluated using exact integer and rational coefficients and Arb ball arithmetic. All 40 values retain at least 500 bits of relative accuracy at 150 working decimal digits and overlap recomputations at 200 digits; 100 significant digits are stored.

A separate continuation of the defining spectral sum expands the multiplicity in the shifted variable $k+(d-1)/2$, treats every pole term analytically, and bounds the remaining tail by a geometric series. Its enclosing calculation at 240 working digits agrees in all dimensions, with truncation error less than $10^{-170}$. The interval control, the negative-even zeta-derivative functional equation, changed tail splits, both convention conversions, all 20 rounded source values and the sphere spectral decomposition were also checked. The generator, independent calculation, source caveat and numerical evidence are attached.