Hearing shapes of drums: Mathematical and physical aspects of isospectrality

In a celebrated paper “Can one hear the shape of a drum?” M. Kac [Am. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality.

Figure 1

Paradigmatic pair of isospectral billiards with seven half-square-shaped base tiles. The dotted lines are just a guide for the eye.

Figure 2

The pair $7_3$ (see Appendix ) of isospectral billiards with a rectangular base shape.

Figure 3

Pictorial representation of the paper-folding method.

Figure 4

Graphs corresponding to a pair of isospectral billiards: If we label the sides of the triangle by $\mu =1,2,3$, the unfolding rule by symmetry with respect to side $\mu$ can be represented by edges made of $\mu$ braids in the graph. From a given pair of graphs, one can construct infinitely many pairs of isospectral billiards by applying the unfolding rules to any shape.

Figure 5

Isospectral billiards. The top left is the seven-edged building block. From 47.

Figure 6

Examples of building blocks yielding isospectral pairs.

Figure 7

The pair $7_3$ of isospectral billiards with a rectangular base shape unfolded to a translation surface (i.e., flat billiard with opposite sides identified).

Figure 8

Two isospectral billiards with a triangular base shape on seven tiles.

Figure 9

The Fano plane.

Figure 10

Periodic and diffractive orbits in the unfolded pair of Fig. 7.

Figure 11

A contribution to the Green function in the case of forward diffraction. If the orbit goes through a vertex, the term in Eq. (74) should be interpreted as the limit for $ϵ\to 0$ of an infinite number of trajectories. If there is no vertex only the straight path contribution remains, the other (winding) terms add up to zero.

Figure 12

A contribution to the Green function in the case of forward diffraction. The filled circles are scattering vertices; the empty ones are nonscattering vertices.

Figure 13

Isospectral billiards divided into smaller regions.

Figure 14

Elementary regions building the isospectral pairs.

Figure 15

Starting half ellipse.

Figure 16

Half ellipse with two bumps.

Figure 17

Perturbed half ellipse with two bumps.

Figure 18

Same as Fig. 17 but perturbed in a dual way.

Figure 19

Isospectral billiards with mixed Neumann-Dirichlet boundary conditions. Solid line, Dirichlet; dashed line, Neumann.

Figure 20

The unique generalized quadrangle of order 2: the symplectic quadrangle $\mathbf{W}(3,2)$.

Figure 21

(Color online) Fundamental mode.

Figure 22

(Color online) First triangular mode (ninth mode).

Figure 23

(Color online) 15th mode.

Figure 24

Pair $7_1$. Sunada triple $G=\mathbf{PSL}(3,2)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(01\right)\left(25\right)$, $b_1=\left(02\right)\left(34\right)$, $c_1=\left(04\right)\left(16\right)$, $a_2=\left(04\right)\left(23\right)$, $b_2=\left(01\right)\left(46\right)$, and $c_2=\left(02\right)\left(15\right)$.

Figure 25

Pair $7_2$. Sunada triple $G=\mathbf{PSL}(3,2)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(01\right)\left(25\right)$, $b_1=\left(15\right)\left(34\right)$, $c_1=\left(04\right)\left(16\right)$, $a_2=\left(04\right)\left(23\right)$, $b_2=\left(06\right)\left(14\right)$, and $c_2=\left(02\right)\left(15\right)$.

Figure 26

Pair $7_3$. Sunada triple $G=\mathbf{PSL}(3,2)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(25\right)\left(46\right)$, $b_1=\left(15\right)\left(34\right)$, $c_1=\left(04\right)\left(16\right)$, $a_2=\left(03\right)\left(24\right)$, $b_2=\left(06\right)\left(14\right)$, and $c_2=\left(02\right)\left(15\right)$.

Figure 27

Pair ${13}_1$. Sunada triple $G=\mathbf{PSL}(3,3)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(012\right)\left(110\right)\left(35\right)\left(67\right)$, $b_1=\left(010\right)\left(29\right)\left(34\right)\left(58\right)$, $c_1=\left(04\right)\left(16\right)\left(211\right)\left(912\right)$, $a_2=\left(04\right)\left(23\right)\left(68\right)\left(910\right)$, $b_2=\left(012\right)\left(14\right)\left(511\right)\left(69\right)$, and $c_2=\left(010\right)\left(15\right)\left(27\right)\left(312\right)$.

Figure 28

Pair ${13}_2$. Sunada triple $G=\mathbf{PSL}(3,3)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(012\right)\left(110\right)\left(35\right)\left(67\right)$, $b_1=\left(112\right)\left(29\right)\left(38\right)\left(45\right)$, $c_1=\left(04\right)\left(16\right)\left(211\right)\left(912\right)$, $a_2=\left(04\right)\left(23\right)\left(68\right)\left(910\right)$, $b_2=\left(01\right)\left(412\right)\left(511\right)\left(810\right)$, and $c_2=\left(010\right)\left(15\right)\left(27\right)\left(312\right)$.

Figure 29

Pair ${13}_3$. Sunada triple $G=\mathbf{PSL}(3,3)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(17\right)\left(35\right)\left(49\right)\left(610\right)$, $b_1=\left(112\right)\left(29\right)\left(38\right)\left(45\right)$, $c_1=\left(04\right)\left(16\right)\left(211\right)\left(912\right)$, $a_2=\left(09\right)\left(410\right)\left(68\right)\left(712\right)$, $b_2=\left(01\right)\left(412\right)\left(511\right)\left(810\right)$, and $c_2=\left(010\right)\left(15\right)\left(27\right)\left(312\right)$.

Figure 30

Pair ${13}_4$. Sunada triple $G=\mathbf{PSL}(3,3)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(17\right)\left(35\right)\left(49\right)\left(610\right)$, $b_1=\left(05\right)\left(12\right)\left(612\right)\left(911\right)$, $c_1=\left(04\right)\left(16\right)\left(211\right)\left(912\right)$, $a_2=\left(09\right)\left(410\right)\left(68\right)\left(712\right)$, $b_2=\left(011\right)\left(18\right)\left(27\right)\left(34\right)$, and $c_2=\left(010\right)\left(12\right)\left(27\right)\left(312\right)$.

Figure 31

Pair ${13}_5$. Sunada triple $G=\mathbf{PSL}(3,3)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(17\right)\left(35\right)\left(49\right)\left(610\right)$, $b_1=\left(05\right)\left(12\right)\left(612\right)\left(911\right)$, $c_1=\left(04\right)\left(16\right)\left(211\right)\left(912\right)$, $a_2=\left(09\right)\left(410\right)\left(68\right)\left(712\right)$, $b_2=\left(011\right)\left(18\right)\left(27\right)\left(34\right)$, and $c_2=\left(010\right)\left(15\right)\left(27\right)\left(312\right)$.

Figure 32

Pair ${13}_6$. Sunada triple $G=\mathbf{PSL}(3,3)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(02\right)\left(17\right)\left(36\right)\left(510\right)$, $b_1=\left(06\right)\left(24\right)\left(38\right)\left(59\right)$, $c_1=\left(05\right)\left(12\right)\left(612\right)\left(911\right)$, $a_2=\left(07\right)\left(311\right)\left(68\right)\left(912\right)$, $b_2=\left(08\right)\left(110\right)\left(511\right)\left(79\right)$, and $c_2=\left(011\right)\left(18\right)\left(27\right)\left(34\right)$.

Figure 33

Pair ${13}_7$. Sun+ada triple $G=\mathbf{PSL}(3,3)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(02\right)\left(17\right)\left(36\right)\left(510\right)$, $b_1=\left(04\right)\left(23\right)\left(68\right)\left(910\right)$, $c_1=\left(05\right)\left(12\right)\left(612\right)\left(911\right)$, $a_2=\left(07\right)\left(311\right)\left(68\right)\left(912\right)$, $b_2=\left(012\right)\left(110\right)\left(35\right)\left(67\right)$, and $c_2=\left(011\right)\left(18\right)\left(27\right)\left(34\right)$.

Figure 34

Pair ${13}_8$. Sunada triple $G=\mathbf{PSL}(3,3)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(010\right)\left(15\right)\left(27\right)\left(312\right)$, $b_1=\left(04\right)\left(23\right)\left(68\right)\left(910\right)$, $c_1=\left(05\right)\left(12\right)\left(612\right)\left(911\right)$, $a_2=\left(04\right)\left(16\right)\left(211\right)\left(912\right)$, $b_2=\left(012\right)\left(110\right)\left(35\right)\left(67\right)$, and $c_2=\left(011\right)\left(18\right)\left(27\right)\left(34\right)$.

Figure 35

Pair ${13}_9$. Sunada triple $G=\mathbf{PSL}(3,3)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(010\right)\left(15\right)\left(27\right)\left(312\right)$, $b_1=\left(110\right)\left(36\right)\left(57\right)\left(911\right)$, $c_1=\left(05\right)\left(12\right)\left(612\right)\left(911\right)$, $a_2=\left(04\right)\left(16\right)\left(211\right)\left(912\right)$, $b_2=\left(03\right)\left(24\right)\left(68\right)\left(711\right)$, and $c_2=\left(011\right)\left(18\right)\left(27\right)\left(34\right)$.

Figure 36

Pair ${15}_1$. Sunada triple $G=\mathbf{PSL}(4,2)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(014\right)\left(112\right)\left(26\right)\left(45\right)\left(711\right)\left(910\right)$, $b_1=\left(113\right)\left(27\right)\left(46\right)\left(89\right)$, $c_1=\left(114\right)\left(212\right)\left(34\right)\left(811\right)$, $a_2=\left(011\right)\left(15\right)\left(34\right)\left(610\right)\left(89\right)\left(1314\right)$, $b_2=\left(010\right)\left(12\right)\left(69\right)\left(1214\right)$, and $c_2=\left(05\right)\left(24\right)\left(67\right)\left(1114\right)$.

Figure 37

Pair ${15}_2$. Sunada triple $G=\mathbf{PSL}(4,2)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(014\right)\left(112\right)\left(26\right)\left(45\right)\left(711\right)\left(910\right)$, $b_1=\left(113\right)\left(27\right)\left(46\right)\left(89\right)$, $c_1=\left(012\right)\left(16\right)\left(35\right)\left(78\right)$, $a_2=\left(011\right)\left(15\right)\left(34\right)\left(610\right)\left(89\right)\left(1314\right)$, $b_2=\left(010\right)\left(12\right)\left(69\right)\left(1214\right)$, and $c_2=\left(013\right)\left(111\right)\left(23\right)\left(710\right)$.

Figure 38

Pair ${15}_3$. Sunada triple $G=\mathbf{PSL}(4,2)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(014\right)\left(211\right)\left(47\right)\left(56\right)\left(810\right)\left(1213\right)$, $b_1=\left(113\right)\left(27\right)\left(46\right)\left(89\right)$, $c_1=\left(012\right)\left(16\right)\left(35\right)\left(78\right)$, $a_2=\left(09\right)\left(25\right)\left(34\right)\left(68\right)\left(1011\right)\left(1213\right)$, $b_2=\left(010\right)\left(12\right)\left(69\right)\left(1214\right)$, and $c_2=\left(013\right)\left(111\right)\left(23\right)\left(710\right)$.

Figure 39

Pair ${15}_4$. Sunada triple $G=\mathbf{PSL}(4,2)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(014\right)\left(211\right)\left(47\right)\left(56\right)\left(810\right)\left(1213\right)$, $b_1=\left(14\right)\left(28\right)\left(79\right)\left(89\right)$, $c_1=\left(012\right)\left(16\right)\left(35\right)\left(78\right)$, $a_2=\left(09\right)\left(25\right)\left(34\right)\left(68\right)\left(1011\right)\left(1213\right)$, $b_2=\left(69\right)\left(713\right)\left(1214\right)$, and $c_2=\left(013\right)\left(111\right)\left(23\right)\left(710\right)$.

Figure 40

Pair ${21}_1$. Sunada triple $G=\mathbf{PSL}(3,4)$, $G_i=⟨a_i,b_i,c_i⟩$, $i=1,2$, with $a_1=\left(27\right)\left(311\right)\left(512\right)\left(818\right)\left(1314\right)\left(1517\right)\left(1620\right)$, $b_1=\left(017\right)\left(38\right)\left(412\right)\left(613\right)\left(919\right)\left(1415\right)\left(1618\right)$, $c_1=\left(18\right)\left(216\right)\left(411\right)\left(519\right)\left(714\right)\left(1017\right)\left(1320\right)$, $a_2=\left(01\right)\left(417\right)\left(712\right)\left(916\right)\left(1020\right)\left(1113\right)\left(1519\right)$, $b_2=\left(020\right)\left(316\right)\left(611\right)\left(815\right)\left(919\right)\left(1012\right)\left(1418\right)$, and $c_2=\left(18\right)\left(216\right)\left(411\right)\left(519\right)\left(714\right)\left(1017\right)\left(1320\right)$.