42
votes
Accepted
Intuitively, what does a graph Laplacian represent?
How to understand the Graph Laplacian (3-steps recipe for the impatients)
read the answer here by Muni Pydi. This is essentially a concentrate of a comprehensive article, which is very nice and well-...
38
votes
Accepted
Spectral symmetry of a certain structured matrix
For real $a,b,c$ and imaginary $d$ the matrix $A$ has chiral symmetry, meaning it anticommutes with a matrix $X$ that squares to the identity:
$$X=\left(
\begin{array}{cccc}
0 & 0 & 0 & -...
37
votes
Can you hear the shape of a drum by choosing where to drum it?
Sometimes you can
Since there seem to be no hard results along these lines on the literature, I decided to have a look at the two shapes in the question and see if there are some relatively ...
32
votes
Accepted
Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?
Preliminary remark. As mentioned in the comments, I find the notion "resolvent formalism", as well as the description in the Wikipedia article, rather misleading - resolvents are not ...
25
votes
Accepted
Can one hear the (topological) shape of a drum?
There are examples due to Ikeda of isospectral Lens spaces which are not homotopy equivalent.
Likeliest the simplest examples are the compact connected 3-dimensional flat manifolds which are a ...
24
votes
Accepted
How are eigenvalues and eigenvectors affected by adding the all-ones matrix?
This is a special case of a rank one perturbation or a rank one update, and there is plenty of work on such. See the nice 2010 lecture notes by Andre Ran.
23
votes
Accepted
Eigenfunctions of the laplacian on $\mathbb{CP}^n$
The $k$-th eigenfunctions are actually easy to describe: In $\mathbb{C}^{n+1}$ with unitary complex coordinates $z_0,z_1,\ldots,z_n$, write $Z = |z_0|^2+\cdots+|z_n|^2$.
Now, for a given $k\ge0$, ...
22
votes
Why is there no symplectic version of spectral geometry?
From a certain point of view the premise of the question is wrong. The study of sympletic manifolds with no additional structure is akin to differential topology rather than differential geometry. ...
20
votes
Accepted
Why is there no symplectic version of spectral geometry?
The characteristic variety (i.e. vanishing locus of the symbol) of a symplectomorphism invariant scalar differential equation is a real projective hypersurface invariant under the group of ...
20
votes
Accepted
Eigenvalue pattern
The explanation is pretty simple with a suitable change of basis.
Letting
$$B =
\begin{pmatrix}
1 & 0 & 1 & 0 \\
i & 0 & -i & 0 \\
0 & 1 & 0 & 1 \\
0 & i &...
19
votes
Accepted
Non real eigenvalues for elliptic equations
Here is a construction. It elaborates from perturbation analysis of eigenvalues. However it starts from the situation of a non-simple eigenvalue.
So, let me start with the standard self-adjoint $L_0=-\...
18
votes
Rigorous justification for this formal solution to $f(x+1)+f(x)=g(x)$
A direct and precise way to arrive at your answer is to appeal to the theory of Fourier transforms of distributions. If I define the Fourier transform as
$$F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\...
17
votes
Accepted
Differentiability of eigenvalues of positive-definite symmetric matrices
In the open subset of $M_n(\mathbb{R})$ where the $\lambda_i$ are distinct, they are $C^{\infty}$ functions: this follows from the implicit function theorem.
On the other hand, when some eigenvalue ...
17
votes
Intuitively, what does a graph Laplacian represent?
This is just a long comment, adding to the excellent answers above.
There is a great article from László Lovász "Discrete and Continuous:
Two sides of the same?", written around 2000 (https:/...
16
votes
Differentiability of eigenvalues of positive-definite symmetric matrices
The keyword is the Cartan decomposition in the theory of symmetric spaces.
In short, when an eigenvalue is simple (its multiplicity is $1$) it is locally an analytic function. But when the ...
16
votes
Spectral symmetry of a certain structured matrix
An equivalent trick : Let $J:= \operatorname{diag}(1,i,-1,-i)$. Then $J^*AJ=iB$ where $B$ is real and skew-symmetric. Hence the spectrum of $iB$ (thus that of $A$) comes by pairs $\pm\lambda$.
15
votes
Accepted
An orbit of symmetric polynomials
Suppose $f(x,y,z)$ is symmetric (in the following, symmetric tout court always means "symmetric w.r.to the three variables $(x,y,z)$") . Then $\mathcal{L}f(x,y,z):=(x+y)(x+z)f(x-1,y,z)-x^2f(x,y,z)$ ...
15
votes
Accepted
Imaginary eigenvalues
Define the unitary and Hermitian matrices
$$U=\left(
\begin{array}{cccc}
0 & 0 & -i & 0 \\
0 & 0 & 0 & -i \\
i & 0 & 0 & 0 \\
0 & i & 0 & 0 \\
\end{...
14
votes
Spectrum of $L^\infty(X,\mu)$
I'm recording here some references on the object $\tilde X = \mathrm{Spec}(L^\infty(X,\Sigma,\mu))$ (many of which were communicated to me recently by Balint Farkas), assuming for sake of simplicity ...
14
votes
Accepted
Eigenvalues of Laplace-Beltrami on half sphere
Using symmetry, you can extend any Dirichlet eigenfunction on the upper half-sphere to the entire sphere $\mathbb{S}^{N-1}$. Therefore, the spectrum of the upper hemisphere is a subset of the spectrum ...
14
votes
Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs?
Perhaps I can contribute to the history part the question, since I was quite close to the Institut Fourier at that time and was very interested in their work (I am a physicist). Grenoble now has ...
14
votes
Intuitively, what does a graph Laplacian represent?
This is not really about the connection with graph theory, a topic I am rather ignorant of, but rather the connection to continuum notions, all of which I learned from this paper.
Consider a ...
14
votes
Accepted
Spectrum of matrix involving quantum harmonic oscillator
The Hamiltonian
$$H=\begin{pmatrix}
\alpha^2+a^\ast a&\alpha a+\beta a^\ast\\
\alpha a^\ast+\beta a&\beta^2+a^\ast a
\end{pmatrix}
$$
is known in the physics literature as the anisotropic Rabi ...
13
votes
Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifold
A modern "simple philosophical" explanation is that this problem can be restated as an eigenvalue problem for a compact operator in an appropriate Hilbert space, whose eigenvalues are reciprocal to ...
13
votes
Differentiability of eigenvalues of positive-definite symmetric matrices
As mentionned by other answers, simple eigenvalues are $C^\infty$, while non-simple ones are not. Let me add however two important properties which you can find in Kato's book Perturbation theory of ...
12
votes
How are eigenvalues and eigenvectors affected by adding the all-ones matrix?
A cute fact that is trivial to prove is this: define the characteristic polynomial of a matrix $M$ by $\phi_M(x) = |xI-M|$. Then for any $A$ and any $s$, $$\phi_{A+sJ}(x) = (1-s)\phi_A(x)+s\phi_{A+J}(...
12
votes
Intuitively, what does a graph Laplacian represent?
I wrote a blog post a while ago* on different ways of interpreting the graph laplacian from the perspectives of functional analysis, probability, statistics, differential equations, and topology, and ...
12
votes
Why is the length spectrum called a spectrum?
I do not know of such an operator. But there is the following lovely theorem of Huber:
Theorem: Two compact hyperbolic surfaces have the same spectrum of the Laplacian if and only if they have the ...
11
votes
Accepted
A $n$-gon is isospectral to a regular $n$-gon (Isospectral $\implies$ isometry ?)
Rowlett is hosting The Sound of Symmetry here. The proof of Theorem 4 is exactly as Noam Elkies suggests: Via the Dirichlet heat trace's asymptotic expansion, both area and perimeter are determined by ...
11
votes
Accepted
Eigenfunction of Laplacian
We can find all the tempered distributions $u$ such that $\Delta u=\lambda u$ (thus, including continuous functions going to $0$ at infinity, since these are locally integrable): taking Fourier ...
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