# Questions tagged [isospectrality]

The isospectrality tag has no usage guidance.

31
questions

3
votes

0
answers

131
views

### Almost conjugate subgroups of compact simple Lie groups

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$Let $G$ be a compact connected Lie group.
Definition:
Two finite subgroups $H_1,H_2$ of $G$ are said to be almost ...

13
votes

1
answer

314
views

### Geometric explanation for coincidence in lengths of 16-dimensional even unimodular lattices?

Question
Up to equivalence, there are two positive-definite even unimodular lattices in $16$ dimensions: $D_{8}^{+}\oplus D_{8}^{+}$ and $D_{16}^{+}$. As observed by Witt in 1941, the theory of ...

1
vote

0
answers

85
views

### A formula involving the heat kernel on the universal cover of a punctured plane

I am looking for the earliest reference to the following formula:
$$
\int_0^\infty\tilde{P}(1,e^{i\alpha},t)\frac{dt}{t}=\frac{1}{\pi \alpha^2},\quad \alpha>0,
$$
where $\tilde{P}(x,y,t)$ is the ...

1
vote

0
answers

201
views

### Calculating frequency of sound of ringing metal coin

I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...

2
votes

0
answers

61
views

### On isospectral planar domains (and a paper by Buser, Conway, Doyle and Semmler)

I have never seen a short, elegant way (from the viewpoint of a non-topologist) which constructs isospectral planar domains from Sunada group triples, although essentially those triples live at the ...

2
votes

0
answers

191
views

### Graphs with the same Laplacian eigenvalues

Let $L$ be the
Laplacian matrix
for a simple graph $G$ of $n$ vertices,
and $\lambda_0,\ldots,\lambda_{n-1}$ its $n$ eigenvalues.
Q.
What is the cardinality of the class of $n$-vertex graphs $\...

4
votes

0
answers

488
views

### Orthogonal similarity of adjacency matrices of graphs which are cospectral and have a common equitable partition

Let $G$ and $H$ be two undirected graphs of the same order (i.e., they have the same number of vertices). Denote by $A_G$ and $A_H$ the corresponding adjacency matrices. Furthermore, denote by $\bar G$...

5
votes

1
answer

334
views

### Poisson summation formula and its implication for the spectrum of the flat torus

I usually ask questions on math.stackexchange but I figure this one is more suited to being asked here. I should preface that I am a complete novice undergraduate, and unlikely to understand answers ...

20
votes

2
answers

2k
views

### Can one hear the (topological) shape of a drum?

Let $(M,g)$ be a (say closed) Riemannian manifold. One can try to understand the geometry/topology of $(M,g)$ by studying the eigenvalues of the Laplacian (this I guess has two versions: when ...

6
votes

2
answers

227
views

### Generalised Isospectrality of Graphs

Q: Is there a graph matrix-representation (not necessarily an $n \times n$ matrix for an $n$-graph) such that isospectrality implies graph-isomorphism? For instance, would the simple distance-matrix ...

5
votes

0
answers

76
views

### Isospectrality, Gassmann-Sunada triples, and tensor products

It is well known that Gassmann-Sunada group triples can be used to construct isospectral manifolds, arithmetically equivalent number fields, etc. (Recall that a Gassmann-Sunada triple $(U,V,W)$ ...

10
votes

3
answers

459
views

### From Gassmann-Sunada triples to isospectral manifolds

A Gassmann-Sunada triple is a triple $(U,V,W)$ of groups, with $V, W$ subgroups of $U$, such that $U$ and $V$ meet every conjugacy class in $U$ in the same number of elements, and such that $V$ and $W$...

1
vote

0
answers

113
views

### The KdV equation is an isospectral evolution of the Schrödinger operator?

I understand that the KdV equation,
$$u_t(x,t)=-u_{xxx}(x,t)-6u(x,t)u_x(x,t)$$
is a compatibility condition of the two equations
$$\psi_{xx}(x,t)=-(u(x,t)+\lambda)\psi(x,t),$$
$$\psi_t(x,t)=u_x(x,...

7
votes

1
answer

421
views

### Generalized Rayleigh-quotient gradient flow on Grassmannian

The following theorem appears without proof in :
Helmke, Uwe, and John B. Moore. Optimization and dynamical systems. Springer Science & Business Media, 2012.
Let $A$ be a symmetric $n\times n$ ...

9
votes

1
answer

408
views

### A $n$-gon is isospectral to a regular $n$-gon (Isospectral $\implies$ isometry ?)

If an $n$-gon $P$ is isospectral to a regular $n$-gon $Q$, what could we say about the shape of the $P$. Otherwise, what could we say about $Q$? In fact, some hints or simply some ideas would be ...

71
votes

2
answers

3k
views

### Can you hear the shape of a drum by choosing where to drum it?

I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian ...

18
votes

3
answers

923
views

### Spectral properties of the Laplace operator and topological properties

Suppose that $M$ is a closed Riemannian manifold: one can construct the so called Laplace-Beltrami operator on $M$. Its spectrum contains some information of the underlying manifold: for example its ...

7
votes

1
answer

427
views

### Progress on isospectral plane domains

Has there been any progress on the smooth isospectral plane domains for Laplacian problem with Dirichlet data? In particular, are there known examples of domains which are isospectral to the unit ...

3
votes

2
answers

921
views

### Can one hear the shape of a drum for operators?

M. Kac in his famous paper "Can one hear the shape of a drum?" asked whether one can "hear" the area of the ambient domain by looking at the spectral picture. Although he was not the first who came up ...

4
votes

2
answers

270
views

### Teichmuller distance between isospectral riemann surfaces

Let $S$ be a surface of negative Euler characteristic (for simplicity let's assume $S$ to be closed), and let $\mathcal{M}(S)$ denote the moduli space of hyperbolic surfaces homeomorphic to $S$.
...

9
votes

1
answer

547
views

### Orders in Central Simple Algebras. Applications

It is known that orders in quaternion algebras (over a number field) are used for constructing geometric objects like hyperbolic orbifolds and Shimura curves. Moreover, if one knows embedding ...

3
votes

1
answer

258
views

### Can we count isospectral graphs?

On n-vertices, how many isospectral graphs exist?
[..I saw this previous "historic" discussion between two of the stalwarts in this field - Operation on Isospectral graphs ]
Given a graph are ...

5
votes

0
answers

212
views

### What do we know about isospectral Cayley graphs?

Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...

12
votes

0
answers

201
views

### Classes for which the Spectrum determines a Convex Shape

Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar ...

4
votes

2
answers

382
views

### are there pairs of combinatorial graphs that are both isospectral and have the same matroid?

Two graphs are isospectral if the combinatorial Laplacian on them has the same spectrum, equivalently, the adjacency matrix has the same the set of eigenvalues (including multiplicities). Two graphs ...

16
votes

1
answer

893
views

### Are isospectral manifolds necessarily homeomorphic?

It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric.
Is it known if there are closed Riemannian manifolds which are isospectral but not homeomorphic?...

3
votes

1
answer

382
views

### Length spectrum and Zoll surfaces of revolution

The earlier MO question, "Length spectrum of spheres," asked if the length spectrum of closed
geodesics determines the metric on $S^2$, and the answer was a clear No due to Zoll surfaces,
all of whose ...

6
votes

2
answers

669
views

### Length spectrum of spheres

It is known (thanks to Hingston, Bangert, Franks, Birckhoff, etc) that $(S^2, g)$ has lots of primitive closed geodesics for any Riemannian metric $g$ (Riemannian is crucial here, this is not true for ...

5
votes

3
answers

375
views

### Operation on Isospectral graphs

Suppose $G$ and $H$ are two isospectral connected graphs. Can we say anything about isospectrality of graphs that are obtained by applying a binary operation to $G$ and $H$?
For example, to take one ...

8
votes

3
answers

894
views

### Classes of graphs for which isospectrum implies isomorphism?

The spectrum of a graph is the (multi)set of eigenvalues of its adjacency matrix (or Laplacian, depending on what you're interested in). In general, two non-isomorphic graphs might have the same ...

8
votes

2
answers

1k
views

### Are vertex and edge-transitive graphs determined by their spectrum?

A graph is called vertex and edge transitive if the automorphism group is transitive on both vertices and edges.
The spectrum of a graph is the collection (with multiplicities) of eigenvalues of the ...