**27**

votes

**3**answers

2k views

### Perron-Frobenius “inverse eigenvalue problem”

The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...

**27**

votes

**6**answers

3k views

### Good references for Rigged Hilbert spaces?

Every now and then I attempt to understand better quantum mechanics and quantum field theory, but for a variety of possible reasons, I find it very difficult to read any kind of physicist account, ...

**25**

votes

**10**answers

5k views

### real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?

**23**

votes

**3**answers

2k views

### Trapped rays bouncing between two convex bodies

At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...

**21**

votes

**4**answers

1k views

### Can $L^{2}$ be represented as a space of functions (not equivalence classes)?

Let $X$ be the vector space of all Lebesgue-measurable functions $f:\left[a,b\right]\rightarrowℝ$ such that $\int^{b}_{a}\left|f\left(x\right)\right|^{2}dx<\infty$ (Lebesgue integral). Then we can ...

**17**

votes

**2**answers

416 views

### Has Witten's perturbation on de Rham complex been studied on other elliptic complexes?

In his famous work, Supersymmetry and Morse theory, Witten perturbs de Rham complex by perturbing the exterior derivative
$$d_h=e^{-ht}de^{ht}.$$
And he proves Morse inequality using some spectral ...

**17**

votes

**1**answer

1k views

### In a Banach algebra, do ab and ba have almost the same exponential spectrum?

Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is ...

**16**

votes

**1**answer

342 views

### Avoiding integers in the spectrum of the Laplacian of a Riemann surface

Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...

**15**

votes

**4**answers

849 views

### Who first used the multiplication operator version of spectral theory

This is another history question.
Hilbert phrased the spectral theorem in terms of resolutions of the identity.
While this remained the form of Stone and von Neumann, they did also have the ...

**15**

votes

**1**answer

2k views

### Fast Fourier Transform for Graph Laplacian?

In the case of a regularly-sampled scalar-valued signal $f$ on the real line, we can construct a discrete linear operator $A$ such that $A(f)$ approximates $\partial^2 f / \partial x^2$. One way to ...

**14**

votes

**2**answers

623 views

### Eigenvalues of an “oblique diagonal” matrix

I am looking for guidance about the behavior of powers of a particular matrix (call it $A_n$ for $n\ge2$), which has come up in a counting problem about quantum knot mosaics (a good reference for ...

**14**

votes

**2**answers

469 views

### Is there a spectral theory approach to non-explicit Plancherel-type theorems?

Teaching graduate analysis has inspired me to think about the completeness theorem for Fourier series and the more difficult Plancherel theorem for the Fourier transform on $\mathbb{R}$. There are ...

**13**

votes

**7**answers

3k views

### Nice applications of the spectral theorem?

Most books and courses on linear algebra or functional analysis present at least one version of the spectral theorem (either in finite or infinite dimension) and emphasize its importance to many ...

**13**

votes

**6**answers

590 views

### Invertibility of a certain matrix indexed by the Hamming cube

For reasons which the margin of this page is too small to hold, I have been reading parts of a recent paper by O. Selim
On submeasures on Boolean algebras, arXiv 1212.6822v3
and in Section 7 the ...

**13**

votes

**4**answers

1k views

### An experiment on random matrices

A bit unsure if my use/mention of proprietary software might be inappropriate or even frowned upon here. If this is the case, or if this kind of experimental question is not welcome, please let me ...

**13**

votes

**4**answers

1k views

### High multiplicity eigenvalue implies symmetry?

It is well known that on any compact Riemannian symmetric space $X$, the eigenvalues of the Laplacian have very high multiplicity (comparable with the Weyl bound), and the resulting actions ...

**13**

votes

**2**answers

1k views

### Minimum off-diagonal elements of a matrix with fixed eigenvalues

Hello,
I am en engineer working in radar research. I came accross a problem I cannot seem to find math literature on it.
I can ask it in two different ways. Perhaps depending on the reader, the ...

**13**

votes

**1**answer

2k views

### Relationship between Green's function and geodesic distance?

I am interested in showing that a certain Green's function can be used to approximate the distance function on a Riemannian manifold in the following sense. Let $(M,g)$ be a Riemannian manifold and ...

**11**

votes

**3**answers

954 views

### What is a “Ramanujan Graph”?

I noticed an apparent conflict in the definition in literature about what is a "Ramanujan graph, which I was wondering if someone could kindly clarify.
(1)
The Hoory-Linial-Wigderson review on ...

**11**

votes

**2**answers

1k views

### Combinatorial proof of (a special case of) the spectral theorem?

The spectral theorem for a real $n \times n$ symmetric matrix $A$ says that $A$ is diagonalizable with all eigenvalues real. If $A$ happens to have non-negative integer entries, it can be interpreted ...

**10**

votes

**8**answers

5k views

### Can a self-adjoint operator have a continuous set of eigenvalues?

This should be a trivial question for mathematicians but not for typical physicists.
I know that the spectrum of a linear operator on a Banach space splits into the so-called "point," "continuous" ...

**10**

votes

**5**answers

922 views

### Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

This question is related to another question, but it is definitely not the same.
Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...

**10**

votes

**2**answers

210 views

### First eigenvalue of the Laplacian on a regular polygon

Consider the Laplacian eigenvalue problem $-\Delta u = \lambda u$ on $\Omega$ with Dirichlet boundary conditions. Let $\lambda_1$ denote the first eigenvalue. The following theorem is well known:
...

**10**

votes

**2**answers

657 views

### Are operators with trivial spectrum nilpotent in a sense?

Being far from analysis, I recently learned about the Invariant subspace problem and came up with the following (perhaps simple or well-known) question.
Let $H$ be a separable complex Hilbert space ...

**10**

votes

**1**answer

349 views

### relationship between eigenvalues of (A-B) and eigenvalues of (A^2-B^2)

Let us suppose that $A_{n}$ and $B_n$ are sequences of positive definite matrices satisfying
$c\leq \lambda_{\min}(A_n)\leq \lambda_{\max}(A_n)\leq C$
and
$c\leq \lambda_{\min}(B_n)\leq ...

**10**

votes

**0**answers

165 views

### Weyl law for Maass forms with nontrivial character

The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...

**9**

votes

**3**answers

2k views

### Number Theory and Geometry/Several Complex Variables

This is a question for all you number theorists out there...based on my skimming of number theory textbooks and survey articles, it seems like most of the applications of geometry and complex ...

**9**

votes

**3**answers

1k views

### The first eigenvalue of the laplacian for complex projective space

What is the exact value of the first eigenvalue of the laplacian for complex projective space viewed as $SU(n+1)/S(U(1)\times U(n))$?

**9**

votes

**4**answers

359 views

### Are these three different notions of a graph Laplacian?

I seem to see three different things that are being called the Laplacian of a graph,
One is the matrix $L_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ ...

**9**

votes

**5**answers

1k views

### Spectral theorem for self-adjoint differential operator on Hilbert space

I need a reference concerning a theorem that shows the following result, stated very roughly:
Given a self-adjoint differential operator densely defined on a Hilbert space, then the given Hilbert ...

**9**

votes

**1**answer

413 views

### Spectrum of Laplacian in non-compact manifolds

What can be said about the spectrum of the Laplace-Beltrami operator on a non-compact, complete Riemannian manifold of finite volume? For example, is the point spectrum non-empty?
What would be a ...

**9**

votes

**1**answer

316 views

### Error in Maurins proof for the nuclear spectral theorem?

I am currently studying the nuclear spectral theorem as presented in K. Maurins Monograph "General Eigenfunction Expansions and Unitary Representations of Topological Groups", second chapter or ...

**9**

votes

**1**answer

219 views

### What happens to continuous spectrum upon discretization?

Excuse me for a bit of an vague question, but I haven't been able to find a definite answer for this for quite some time. My question is regarding (mostly non-normal )linear operators and their ...

**9**

votes

**3**answers

422 views

### Why is this operator compact?

Let $D$ be the Dirac-Operator on $\mathbb{R}^n$ or more generally the Dirac spinor bundle $\mathcal{S}\to M$ of a (semi-)Riemannian spin manifold $M$. Then we consider $D$ as an unbouded Operator on ...

**9**

votes

**2**answers

1k views

### What is a good reference that compact resolvent implies Fredholm operator?

Suppose $A \in \mathcal{L}(E_1, E_0)$ is a bounded linear operator between Banach spaces $E_1$ and $E_0$, and we also have that $E_1$ is densely, continuously embedded in $E_0$ (i.e. $A$ can be ...

**8**

votes

**3**answers

296 views

### Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$

Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D ...

**8**

votes

**2**answers

490 views

### Efficiently computing a few localized eigenvectors

Let $H = \triangle + V(x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am interested in domain decomposition for an eigenproblem involving $H$.
The lowest 1000 eigenfunctions of $H$, $ \psi_i $, can ...

**8**

votes

**3**answers

644 views

### Functions of Pseudodifferential Operators

Suppose I have a self-adjoint pseudo-differential operator $A$ on $\mathbb{R}^n$ and a continuous function $f$ (possibly bounded, or Schwartz, or compactly supported) on its spectrum. Then I can ...

**8**

votes

**3**answers

289 views

### Spectrum of Dirichlet Laplacian on a Parallelogram

Let $ M \subset \mathbb{R}^2 $ be parallelogram constructed by putting together two equilateral triangles (so that all sides of the parallelogram have length 1, and the internal angles are 60 and ...

**8**

votes

**1**answer

135 views

### Can the graph Laplacian be well approximated by a Laplace-Beltrami operator?

It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L_{G}$ for some $G$ (where $G$ is ...

**8**

votes

**1**answer

723 views

### Perturbations of an operator that disconnect the spectrum

The following question came to me while working on a technical matter about transversality in infinite dimension, and I'm really curious to know whether it has an affirmative answer at least under ...

**8**

votes

**1**answer

481 views

### A variation on “Hearing the shape of a drum” for polytopes.

Let $\varphi:\mathcal S^{d-1}\longrightarrow \mathbb R_{>0}$ be a strictly positive function describing the boundary $\varphi(\mathbf x)\mathbf x,\mathbf x\in\mathbb S^{d-1}$ of a $d-$dimensional ...

**8**

votes

**4**answers

573 views

### Does a quantitative version of Fredholm theory exist?

Let $X$ be a Banach space and $K:X\to X$ be a compact operator. If $I+K$ is injective then it is onto and hence the inverse $(I+K)^{-1}$ is bounded. What kind of qualitative or quantitative ...

**8**

votes

**1**answer

553 views

### $Spin^c$-Dirac-operator on the 3-torus

Consider the spinc structure on the flat standard 3-torus, which you get from the trivial (or any other) spin structure. Its associated vector bundle can be identified with a trivial bundle with fibre ...

**8**

votes

**0**answers

485 views

### Bounding sum of first singular values squared for Kronecker sum of traceless matrices

Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e.
$$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + ...

**7**

votes

**3**answers

421 views

### Is there a continuous analogue of Ramanujan graphs?

I think it might help to think of the following definition of a Ramanujan graph - a graph whose non-trivial eigenvalues are such that their magnitude is bounded above by the spectral radius of its ...

**7**

votes

**2**answers

710 views

### Resolvent of Laplacian

Hello!
Let $(M,g)$ be a Riemannian manifold and $-\Delta$ the Laplacian on M (acting on smooth functions). Then the resolvent $R(\xi)$ of $-\Delta$ is a compact operator.
Is it possible to find for ...

**7**

votes

**2**answers

544 views

### What are first eigenfunctions of Laplacian for $CP^n$ with Fubini-Study metric?

I know the round $n$-sphere has $f_i=\cos(dist(e_i, x))$ as the set of first eigenfunctions for $e_i=(0, \cdots, 1, \cdots, 0)\in \mathbb R^{n+1}$. i.e. $\Delta f_i=\lambda_1 f$, where $\lambda_1$ is ...

**7**

votes

**7**answers

921 views

### What is the best reference for Spectral theory?

I'm studying Bernard Aupetit: A Primer on Spectral Theory
but the textbook we are using is a little bit heavy going for me. Is there a best book to learn about these things?
Thank you.

**7**

votes

**1**answer

247 views

### Eigenfunctions restricted on closed geodesics

Consider the flat torus $T^2=\frac{\mathbb{R}^2}{l_1\mathbb{Z}\oplus l_2\mathbb{Z}}$. It is easy to see that the eigenvalues of the Laplacian on torus, $-\frac{\partial^2}{\partial ...