Eigenfunction database (β)

Basic — individual domains

Unit square#square

[0,1] × [0,1]. Separable; every eigenvalue is π²(m²+n²) with m,n ≥ 1.

Ref. Courant–Hilbert (1953), §V.5

mesh: 513 vertices, 944 triangles, P2 Lagrange

Dirichlet boundary

Unit square, dirichlet, focused mode — k = 1 λ = 19.739246

Neumann boundary

Unit square, neumann, focused mode — k = 1 λ = 0

Robin boundary

α = 1 2 / 4

Unit square, robin, focused mode — k = 1 λ = 3.414106

Steklov boundary

Unit square, steklov, focused mode — k = 1 λ = 0

Unit disk#disk

Radius 1. Eigenvalues are squares of zeros of Bessel J_m.

Ref. Courant–Hilbert (1953), §V.5; Girouard–Polterovich, J. Spectral Theory 7 (2017).

mesh: 1,594 vertices, 3,058 triangles, P2 Lagrange

Dirichlet boundary

Unit disk, dirichlet, focused mode — k = 1 λ = 5.785540

Neumann boundary

Unit disk, neumann, focused mode — k = 1 λ = 0

Robin boundary

α = 1 2 / 4

Unit disk, robin, focused mode — k = 1 λ = 1.577503

Steklov boundary

Unit disk, steklov, focused mode — k = 1 λ = 0

Polygonal — individual domains

Equilateral triangle#equilateral-triangle

Side 1. Lamé (1833) derived the closed-form spectrum λ_{m,n} = (16π²/9)(m² + n² + mn).

Ref. Lamé (1833); McCartin, SIAM Rev. 45 (2003).

mesh: 231 vertices, 400 triangles, P2 Lagrange

Dirichlet boundary

Equilateral triangle, dirichlet, focused mode — k = 1 λ = 52.638598

Neumann boundary

Equilateral triangle, neumann, focused mode — k = 1 λ = 0

Robin boundary

α = 1 2 / 4

Equilateral triangle, robin, focused mode — k = 1 λ = 6.073964

Steklov boundary

Equilateral triangle, steklov, focused mode — k = 1 λ = 0

Right isoceles triangle#right-isoceles-triangle

Legs of length 1. Half-square: Dirichlet spectrum is π²(m²+n²) with m > n ≥ 1.

mesh: 279 vertices, 487 triangles, P2 Lagrange

Dirichlet boundary

Right isoceles triangle, dirichlet, focused mode — k = 1 λ = 49.348577

Neumann boundary

Right isoceles triangle, neumann, focused mode — k = 1 λ = 0

Robin boundary

α = 1 2 / 4

Right isoceles triangle, robin, focused mode — k = 1 λ = 5.814958

Steklov boundary

Right isoceles triangle, steklov, focused mode — k = 1 λ = 0

30-60-90 triangle#30-60-90-triangle

Short leg 1, long leg √3. A tiling (Lamé) triangle with fully integrable billiard.

Ref. Integrable triangle (30-60-90 is one of the three Lamé triangles).

mesh: 456 vertices, 815 triangles, P2 Lagrange

Dirichlet boundary

30-60-90 triangle, dirichlet, focused mode — k = 1 λ = 30.705572

Neumann boundary

30-60-90 triangle, neumann, focused mode — k = 1 λ = 0

Robin boundary

α = 1 2 / 4

30-60-90 triangle, robin, focused mode — k = 1 λ = 4.365073

Steklov boundary

30-60-90 triangle, steklov, focused mode — k = 1 λ = 0

Isoceles trapezoid#trapezoid-iso

Isoceles trapezoid with parallel bases 2 and 1, height 1.

mesh: 775 vertices, 1,442 triangles, P2 Lagrange

Dirichlet boundary

Isoceles trapezoid, dirichlet, focused mode — k = 1 λ = 14.751591

Neumann boundary

Isoceles trapezoid, neumann, focused mode — k = 1 λ = 0

Robin boundary

α = 1 2 / 4

Isoceles trapezoid, robin, focused mode — k = 1 λ = 2.838014

Steklov boundary

Isoceles trapezoid, steklov, focused mode — k = 1 λ = 0

Non-convex — individual domains

L-shape#l-shape

[-1,1]² with the lower-right quadrant removed. The eigenfunctions have an r^{2/3} corner singularity at the re-entrant corner (Fox–Henrici–Moler 1967).

Ref. Fox–Henrici–Moler, SIAM J. Numer. Anal. 4 (1967).

mesh: 1,486 vertices, 2,810 triangles, P2 Lagrange

Dirichlet boundary

L-shape, dirichlet, focused mode — k = 1 λ = 9.645378

Neumann boundary

L-shape, neumann, focused mode — k = 1 λ = 0

Robin boundary

α = 1 2 / 4

L-shape, robin, focused mode — k = 1 λ = 2.100698

Steklov boundary

L-shape, steklov, focused mode — k = 1 λ = 0

Non-concentric annulus#annulus-offcenter

Unit disk with an interior disk of radius 0.25 centred at (0.4, 0) removed. Symmetry-broken variant of the annulus.

mesh: 4,062 vertices, 7,868 triangles, P2 Lagrange

Dirichlet boundary

Non-concentric annulus, dirichlet, focused mode — k = 1 λ = 9.938295

Neumann boundary

Non-concentric annulus, neumann, focused mode — k = 1 λ = 0

Robin boundary

α = 1 2 / 4

Non-concentric annulus, robin, focused mode — k = 1 λ = 2.133803

Steklov boundary

Non-concentric annulus, steklov, focused mode — k = 1 λ = 0

Curved boundary — individual domains

Reuleaux triangle#reuleaux-triangle

Intersection of three unit disks centred at the vertices of an equilateral triangle of side 1; a curve of constant width (Reuleaux 1875).

mesh: 4,801 vertices, 9,363 triangles, P2 Lagrange

Dirichlet boundary

Reuleaux triangle, dirichlet, focused mode — k = 1 λ = 27.275313

Neumann boundary

Reuleaux triangle, neumann, focused mode — k = 1 λ = 0

Robin boundary

α = 1 2 / 4

Reuleaux triangle, robin, focused mode — k = 1 λ = 3.948207

Steklov boundary

Reuleaux triangle, steklov, focused mode — k = 1 λ = 0

Chaotic billiards — individual domains

Cardioid#cardioid

r = 1 − cos θ. The classical billiard is ergodic, mixing and K (Markarian 1993); extensively studied in quantum chaos (Robnik 1983; Bäcker–Steiner 1998).

Ref. Robnik, J. Phys. A 16 (1983); Bäcker–Steiner (1998).

mesh: 7,387 vertices, 14,532 triangles, P2 Lagrange

Dirichlet boundary

Cardioid, dirichlet, focused mode — k = 1 λ = 4.043005

Neumann boundary

Cardioid, neumann, focused mode — k = 1 λ = 0

Robin boundary

α = 1 2 / 4

Cardioid, robin, focused mode — k = 1 λ = 1.273208

Steklov boundary

Cardioid, steklov, focused mode — k = 1 λ = 0

Polygonal — parametric families

Regular n-gon#regpoly

Regular n-gon inscribed in the unit circle. As n → ∞, the spectrum converges to that of the unit disk.

n = 3 1 / 298

Dirichlet boundary

Regular n-gon, dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Regular n-gon, neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Regular n-gon, robin, focused mode — k = 1 λ = —

Steklov boundary

Regular n-gon, steklov, focused mode — k = 1 λ = —

Isoceles triangle (apex angle θ)#isotri

Isoceles triangle with base 1 and apex angle θ. At θ = 60° reduces to the equilateral triangle.

θ (deg) = 10 1 / 55

Dirichlet boundary

Isoceles triangle (apex angle θ), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Isoceles triangle (apex angle θ), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Isoceles triangle (apex angle θ), robin, focused mode — k = 1 λ = —

Steklov boundary

Isoceles triangle (apex angle θ), steklov, focused mode — k = 1 λ = —

Rhombus (acute angle θ)#rhombus

Unit-side rhombus with acute interior angle θ. At θ = 90° reduces to a unit square.

θ (deg) = 5 1 / 43

Dirichlet boundary

Rhombus (acute angle θ), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Rhombus (acute angle θ), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Rhombus (acute angle θ), robin, focused mode — k = 1 λ = —

Steklov boundary

Rhombus (acute angle θ), steklov, focused mode — k = 1 λ = —

Non-convex — parametric families

Annulus (inner radius r)#annulus

Concentric annulus { r < |x| < 1 }. Eigenvalues are determined by cross-products of Bessel functions J_m, Y_m.

r = 0.05 1 / 19

Dirichlet boundary

Annulus (inner radius r), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Annulus (inner radius r), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Annulus (inner radius r), robin, focused mode — k = 1 λ = —

Steklov boundary

Annulus (inner radius r), steklov, focused mode — k = 1 λ = —

Pac-Man (opening θ removed)#pacman

Unit disk with a sector of opening angle θ removed; the re-entrant angle at the origin is 2π − θ, generating an r^{π/(2π−θ)} corner singularity.

θ (rad) = 0.1309 1 / 22

Dirichlet boundary

Pac-Man (opening θ removed), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Pac-Man (opening θ removed), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Pac-Man (opening θ removed), robin, focused mode — k = 1 λ = —

Steklov boundary

Pac-Man (opening θ removed), steklov, focused mode — k = 1 λ = —

Curved boundary — parametric families

Rectangle (aspect ratio r)#rect

Rectangle [0, r] × [0, 1]. Separable spectrum λ_{m,n} = π²((m/r)² + n²).

r = 1 1 / 28

Dirichlet boundary

Rectangle (aspect ratio r), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Rectangle (aspect ratio r), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Rectangle (aspect ratio r), robin, focused mode — k = 1 λ = —

Steklov boundary

Rectangle (aspect ratio r), steklov, focused mode — k = 1 λ = —

Ellipse (aspect ratio e)#ellipse

Ellipse with semi-axes (1, 1/e). Eigenfunctions are products of angular and radial Mathieu functions.

e = 1.1 1 / 32

Dirichlet boundary

Ellipse (aspect ratio e), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Ellipse (aspect ratio e), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Ellipse (aspect ratio e), robin, focused mode — k = 1 λ = —

Steklov boundary

Ellipse (aspect ratio e), steklov, focused mode — k = 1 λ = —

Sectors — parametric families

Circular sector (opening α)#sector

Unit disk sector of opening α. The eigenfunctions are J_{kπ/α}(√λ r) sin(kπθ/α).

α (rad) = 0.1309 1 / 47

Dirichlet boundary

Circular sector (opening α), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Circular sector (opening α), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Circular sector (opening α), robin, focused mode — k = 1 λ = —

Steklov boundary

Circular sector (opening α), steklov, focused mode — k = 1 λ = —

Chaotic billiards — parametric families

Robnik billiard (ε)#robnik

Image of the unit disk under the conformal map z = w + εw². Interpolates between the integrable disk (ε = 0) and increasingly chaotic billiards; ε = 1/2 yields the cardioid.

Ref. Robnik, J. Phys. A 16 (1983).

ε = 0 1 / 25

Dirichlet boundary

Robnik billiard (ε), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Robnik billiard (ε), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Robnik billiard (ε), robin, focused mode — k = 1 λ = —

Steklov boundary

Robnik billiard (ε), steklov, focused mode — k = 1 λ = —

Sinai billiard (hole radius r)#sinai-family

Square [-1,1]² with a central disk of radius r removed. The classical billiard is a K-system (Sinai 1970) for 0 < r < 1.

Ref. Sinai, Russ. Math. Surveys 25 (1970).

r = 0.05 1 / 18

Dirichlet boundary

Sinai billiard (hole radius r), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Sinai billiard (hole radius r), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Sinai billiard (hole radius r), robin, focused mode — k = 1 λ = —

Steklov boundary

Sinai billiard (hole radius r), steklov, focused mode — k = 1 λ = —

Stadium (half-length a)#stadium-family

Rectangle [-a, a] × [-1, 1] with unit semicircular caps. The classical billiard is ergodic and K-mixing for every a > 0 (Bunimovich 1974); a = 0 is the unit disk.

Ref. Bunimovich, Commun. Math. Phys. 65 (1979).

a = 0.1 1 / 25

Dirichlet boundary

Stadium (half-length a), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Stadium (half-length a), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Stadium (half-length a), robin, focused mode — k = 1 λ = —

Steklov boundary

Stadium (half-length a), steklov, focused mode — k = 1 λ = —

Collapsing / thin — parametric families

Isoceles triangle (height h)#thintri

Isoceles triangle with base 1 and height h. The limit h → 0 realises the thin-domain asymptotics λ ~ (const)/h² (Friedlander–Solomyak 2009).

Ref. Friedlander–Solomyak, ESAIM COCV (2009).

h = 0.02 1 / 20

Dirichlet boundary

Isoceles triangle (height h), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Isoceles triangle (height h), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Isoceles triangle (height h), robin, focused mode — k = 1 λ = —

Steklov boundary

Isoceles triangle (height h), steklov, focused mode — k = 1 λ = —

Dumbbell (neck half-width n)#dumbbell

Two unit disks centred at (±1.3, 0) connected by a rectangular channel of half-width n. As n → 0 the first two Dirichlet eigenvalues become asymptotically degenerate (Jimbo 1989; Arrieta 1995).

Ref. Jimbo, J. Diff. Eq. 77 (1989); Arrieta, J. Diff. Eq. 118 (1995).

n = 0.02 1 / 25

Dirichlet boundary

Dumbbell (neck half-width n), dirichlet, focused mode — k = 1 λ = —

Neumann boundary

Dumbbell (neck half-width n), neumann, focused mode — k = 1 λ = —

Robin boundary

α = 1 1 / 1

Dumbbell (neck half-width n), robin, focused mode — k = 1 λ = —

Steklov boundary

Dumbbell (neck half-width n), steklov, focused mode — k = 1 λ = —

3D domains — parametric families

Ball B³ (radius R)#ball

Open 3-ball of radius R with the Dirichlet Laplacian. Eigenvalues are squared zeros of spherical Bessel j_l; shown as a rotating GIF with a transparent hemisphere and a coloured cut-plane.

R = 0.5 1 / 10