Dirichlet eigenvalues of the classical planar domains

edit · history · discussion · files · long url · analysis eigenvalue

Numbers

$\Omega$

$n$

$\lambda_n(\Omega)$

unit square

19.7392088021787172376689819998

$1\times2$ rectangle

12.3370055013616982735431137498

unit disk

5.78318596294678452117599575846

half disk

14.6819706421238932572197777686

quarter disk

26.3746164271633907701130803553

annulus of radii $1$ and $2$

9.75332212475071491068952390873

circular sector of angle $\pi/3$

40.7064658182003197420524327044

equilateral triangle

52.6378901391432459671172853327

right-isosceles triangle

49.3480220054467930941724549994

hemiequilateral triangle

122.821743658000907256606999110

L-shaped membrane

9.63972384402194105271145926236

first Gordon-Webb-Wolpert drum

2.53794399980

comment: The first Gordon-Webb-Wolpert drum is isospectral to the second.

second Gordon-Webb-Wolpert drum

2.53794399980

comment: This equals the corresponding entry for the first Gordon-Webb-Wolpert drum.
equals: the first GWW drum entry

regular pentagon

7.95708938934945889838046059041

regular hexagon

7.15533913392605512821001761683

regular heptagon

6.73509919668492378154956248625

regular octagon

6.48493349372291109071821798842

Definition

For a bounded planar domain $\Omega$, this table gives the Dirichlet eigenvalues $\lambda_n(\Omega)$ of $-\Delta u=\lambda u$ on $\Omega$ with $u=0$ on $\partial\Omega$, listed in increasing order. The index $n$ starts at $1$ and repeats eigenvalues according to multiplicity.

Parameters

$\Omega$

— domain ($\Omega$ is one of the listed classical planar domains.)

$n$

— eigenvalue index ($n\geq1$)

Formulas

(1)

For a rectangle $[0,a]\times[0,b]$, the Dirichlet eigenvalues are $\pi^2(j^2/a^2+k^2/b^2)$ with $j,k\geq1$.

(2)

For a disk or circular sector of radius $1$, separation of variables gives eigenvalues $j_{\nu,k}^2$, where $j_{\nu,k}$ is the $k$th positive zero of $J_\nu$. The unit disk uses $\nu=0,1,2,\ldots$, with multiplicity two for $\nu>0$; a sector of angle $\alpha$ uses $\nu=m\pi/\alpha$ for $m\geq1$.

(3)

The annulus entries are the squares of the positive roots of $J_m(\kappa)Y_m(2\kappa)-J_m(2\kappa)Y_m(\kappa)=0$, with $\kappa>0$, $m=0,1,2,\ldots$ and multiplicity two for $m>0$.

(4)

For the unit-side equilateral triangle, the Dirichlet eigenvalues are $\frac{16\pi^2}{9}(j^2+jk+k^2)$ with $j,k\geq1$ [4].

(5)

For the right-isosceles triangle with legs $1$, the Dirichlet eigenvalues are $\pi^2(j^2+k^2)$ with $1\leq j<k$.

(6)

For the hemiequilateral triangle with hypotenuse $1$, the Dirichlet eigenvalues are $\frac{16\pi^2}{9}(j^2+jk+k^2)$ with $j>k\geq1$ [4].

Comments

(7)

The sign convention is $-\Delta u=\lambda u$, so the Dirichlet eigenvalues are positive. The value stored is $\lambda_n(\Omega)$, not $\sqrt{\lambda_n(\Omega)}$.

(8)

The domains are not rescaled. The unit square is $[0,1]^2$, the $1\times2$ rectangle is $[0,1]\times[0,2]$, the unit disk and the circular sectors have radius $1$, and the annulus is $\{(x,y):1<x^2+y^2<4\}$. The half disk is the sector of angle $\pi$, the quarter disk is the sector of angle $\pi/2$, and the 60-degree sector has angle $\pi/3$.

(9)

The equilateral triangle has vertices $(0,0)$, $(1,0)$ and $(1/2,\sqrt3/2)$. The right-isosceles triangle has vertices $(0,0)$, $(1,0)$ and $(0,1)$. The hemiequilateral triangle is the half of the unit-side equilateral triangle with vertices $(0,0)$, $(1/2,0)$ and $(1/2,\sqrt3/2)$.

(10)

The regular polygons have circumradius $1$. Each is centered at the origin and has a vertex at $(1,0)$.

(11)

The L-shape is $[-1,1]^2\setminus((0,1]\times[-1,0])$, the MATLAB logo domain [5], made from three unit squares.

(12)

The Gordon-Webb-Wolpert drums are the two eight-sided polygons used in [6]. The first Gordon-Webb-Wolpert drum has vertices $(-3,-3)$, $(-3,-1)$, $(1,3)$, $(1,1)$, $(3,1)$, $(1,-1)$, $(-1,-1)$ and $(-1,-3)$. The second Gordon-Webb-Wolpert drum has vertices $(-3,1)$, $(1,1)$, $(1,3)$, $(3,1)$, $(1,-1)$, $(-1,-1)$, $(-1,-3)$ and $(-3,-1)$.

Programs

(P1)

Sage

import mpmath as mp

mp.mp.dps = 50

# These formulas cover the rectangle, disk, sector, annulus, and triangle rows.
def annulus_root():
    def determinant(x):
        return (mp.besselj(0, x) * mp.bessely(0, 2*x)
                - mp.besselj(0, 2*x) * mp.bessely(0, x))
    left = mp.mpf("3.0")
    right = mp.mpf("3.2")
    f_left = determinant(left)
    f_right = determinant(right)
    if f_left * f_right >= 0:
        raise ValueError("the annulus bracket does not change sign")
    for _ in range(250):
        middle = (left + right) / 2
        f_middle = determinant(middle)
        if f_left * f_middle <= 0:
            right = middle
            f_right = f_middle
        else:
            left = middle
            f_left = f_middle
    return (left + right) / 2

values = [
    ("unit square", 2 * mp.pi**2),
    ("1 x 2 rectangle", 5 * mp.pi**2 / 4),
    ("unit disk", mp.besseljzero(0, 1)**2),
    ("half disk", mp.besseljzero(1, 1)**2),
    ("quarter disk", mp.besseljzero(2, 1)**2),
    ("annulus of radii 1 and 2", annulus_root()**2),
    ("sector of angle pi/3", mp.besseljzero(3, 1)**2),
    ("equilateral triangle", 16 * mp.pi**2 / 3),
    ("right-isosceles triangle", 5 * mp.pi**2),
    ("hemiequilateral triangle", 112 * mp.pi**2 / 9),
]
for name, value in values:
    print(name, mp.nstr(value, 32))

References

[1]

Robert S. Jones, Computing ultra-precise eigenvalues of the Laplacian within polygons, 2016. (arXiv)

[2]

Tobin A. Driscoll, Eigenmodes of isospectral drums, SIAM Review 39 (1997), 1-17. (doi)

[3]

Lloyd N. Trefethen and Timo Betcke, Computed eigenmodes of planar regions, Contemporary Mathematics 412 (2006), 297-314. (doi)

[4]

Joseph F. Breen and David Sher, Spectral invariants of integrable polygons, Journal of Fourier Analysis and Applications 31 (2025). (doi)

Links

[5]

MathWorks: Eigenvalues and Eigenmodes of L-Shaped Membrane

[6]

COMSOL: Isospectral Drums

[7]

Toby Driscoll: Isospectral drums

Similar tables

Zeros of Bessel functions of the first kind $J_\alpha$ — the disk and sector eigenvalues are squares of Bessel-zero entries

Local extrema of Bessel functions of the first kind $J_\alpha$ — the extrema of $J_\alpha$ give Neumann disk eigenvalues, in parallel with the Bessel zeros used here for Dirichlet disk and sector eigenvalues

Data properties

Entries are of type: real number

How well the digits are known: heuristic

Table is complete: no (it holds the first Dirichlet eigenvalue for each listed domain)

How they were obtained:

The rectangle, disk, circular-sector, annulus and triangle rows are computed from the displayed formulas at two working precisions and compared. The L-shape value is transcribed from [1]'s hundred-digit computation and checked against the first value reported by [3].

The Gordon-Webb-Wolpert value is transcribed from [2]'s twelve-digit table and stored for both isospectral drums. The regular-polygon rows are transcribed from [1]'s area-$\pi$ values and rescaled to circumradius $1$; these six single-source transcriptions are not agreement checks.