**8**

votes

**1**answer

127 views

### The scope of correspondence principle in quantum chaos

My understanding of the so-called correspondence principle in quantum chaos, is that it is a connection between the behaviour of a classical Hamiltonian system (chaotic/completely integrable) and the ...

**11**

votes

**1**answer

703 views

### Reasons for difficulty of Graph Isomorphism and why Johnson graphs are important?

In http://jeremykun.com/2015/11/12/a-quasipolynomial-time-algorithm-for-graph-isomorphism-the-details/ it is mentioned:
'In discussing Johnson graphs, Babai said they were a source of “unspeakable ...

**4**

votes

**3**answers

191 views

### Show that the Laplacian operator on the Heisenberg group is negative

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law
$$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$
For $z=x+ i y \in \mathbb C$ ...

**1**

vote

**1**answer

50 views

### Eigenfunction of an uniformly elliptic second order operator

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^d$ and let $L$ be a uniformly elliptic second order partial differential operator:
$$Lu(x,t)=-\sum_{i,j=1}^{d}{a_{ij}(x,t)u_{x_{i}x_{j}}(x,t)}+\...

**3**

votes

**0**answers

248 views

### Isolated Eigenvalue of $T$ is also an isolated eigenvalue of $\overline{T}$?

I am working with transfer operators and I reach a point where would be nice if I could use a result from Tosio Kato's book about perturbation theory of linear operators. I think I am able to use Kato'...

**0**

votes

**0**answers

81 views

### Spectrum of the Grassmannian Laplacian

The spectrum of the Laplacian (with respect to the Fubini--Study metric) was addressed in this old question. Does anyone know if these results have been extended to the Grassmannians?

**4**

votes

**1**answer

88 views

### Commutator representation of certain smoothing operators

I have a question regarding the classical trace $\text{Tr} \colon \Psi^{-\infty}(S^1)\to \mathbb C$ on pseudodifferential operators of infinite negative order (i.e. smoothing operators), defined over ...

**1**

vote

**0**answers

40 views

### Fundamental gap for Neumann BVP with potential

I am sure this is extremely well known but I have been digging a bit and I can't find what I need. Consider $B$ to be the unit ball in $ \mathbb R^N$ and consider the eigenvalue problem
\begin{cases}...

**4**

votes

**2**answers

163 views

### Non-asympototic version of Gelfand's formula

Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.
There exists universal ...

**3**

votes

**0**answers

93 views

### Spectral theory of differential forms over a circle bundle

Here is the set up : I consider the unitary tangent bundle of a surface $(S,g)$ endowed with the Sasakian metric ; $(T^1S, g_s)$, in fact we have the following fibration :
\begin{equation*}
\mathbb{...

**10**

votes

**2**answers

217 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 ...

**3**

votes

**1**answer

105 views

### Horn's spectrum problem with random Hermitian matrices

An important problem in matrix analysis, completely solved in the early 2000's by A. Knutson & T. Tao (The honeycomb model of GLn(C) tensor products. I. Proof of the
saturation conjecture. J. Amer....

**4**

votes

**1**answer

180 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 ...

**6**

votes

**0**answers

149 views

### Two proofs of the Cheeger-Müller theorem

In the late 1970's, Cheeger and Müller independently proved the equality of analytic torsion and Reidemeister torsion for orthogonal representations, which had been conjectured by Ray-Singer. Their ...

**3**

votes

**0**answers

73 views

### Spectral theory of Bochner integral operators

Consider the following (somewhat simplified) situation. Let $\mathcal{H}$ be separable Hilbert space and $\mathcal{B}(\mathcal{H})$ the Banach algebra of bounded linear operators acting on $\mathcal{H}...

**9**

votes

**1**answer

366 views

### Multiplicity of Laplace eigenvalues

Disclaimer: This is a very heuristic question and I will be satisfied with heuristic insights, if rigorous and precise answers are not possible.
All the examples of closed surfaces (or higher ...

**3**

votes

**0**answers

47 views

### Length and laplacian spectrum for quasi-fuchsian manifold

It is well known that, in the case of finite area hyperbolic surfaces, the length sprectrum (the collection of length of all closed geodesics) and the spectrum of the laplacian (acting on functions) ...

**5**

votes

**1**answer

227 views

### Convergence of Riemannian metrics spectra

Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...

**8**

votes

**0**answers

225 views

### A question on a result of Colin de Verdiere

Consider a compact connected surface $M$ of some genus $\gamma \geq 2$. A particular case of a famous result of Colin de Verdiere (see here) says that if we fix $\gamma$ and select a finite sequence $\...

**6**

votes

**1**answer

183 views

### History of spectral methods to the study of real analytic $GL_2$-Eisenstein series

I'm trying to sort out the history of spectral methods in the study of real analytic $GL_2$-Eisenstein series. From what I read so far, I would say that the subject was really kicked off by the ...

**4**

votes

**1**answer

147 views

### Is the sum of spectral projections a projection?

Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections
$$P_{\{\lambda_1,...\lambda_n\}}=\frac{...

**3**

votes

**3**answers

202 views

### Compact surfaces with arbitrary gaps in spectrum

Consider a sequence of positive numbers $a_n$. My question is, can we select a closed Riemann surface whose spectrum $\lambda_i$ satisfies the condition that $\lambda_{i + 1} - \lambda_i > a_i$? Of ...

**4**

votes

**0**answers

165 views

### Comparing spectra of Laplacian and Schrödinger operator

Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\...

**2**

votes

**1**answer

98 views

### Spectral geometry: asymptotic sequences of subspaces of $L^2(M)$ and the geometry of $M$

Consider a closed connected Riemannian manifold $M$, together with the associated Hilbert space $L^2(M)$ defined with respect to the Riemannian volume density. Let $-\Delta$ be the positive Laplacian $...

**9**

votes

**1**answer

226 views

### $C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...

**2**

votes

**0**answers

80 views

### About optimizing a convex function on a hypercube

Given a real valued convex function $g$ on $[-1,1]^n$, let $f$ be the restriction of it on the hypercube $\{-1,1\}^n$. I want to find a vertex on the hypercube $\{-1,1\}^n$ on which either (1) $f$ ...

**1**

vote

**0**answers

80 views

### Lp norm estimates for the inverse of the Laplacian on a graph

I am looking at a finite connected graph and I would like to know what is the best [i.e. largest] constant $\lambda_p$ in
$$
\sum_x f(x) =0 \implies \| \Delta^{-1} f\|_{\ell^p} \leq \lambda_p^{-1} \|...

**0**

votes

**2**answers

133 views

### Estimating the shift in the $\lambda_{max}$ of a matrix under a diagonal perturbation

Given a matrix $A$ and a diagonal matrix $D$, what ways do we have to estimate, $\lambda_{max}(A+D) - \lambda_{max}(A)$? (Feel free to make other assumptions about the matrices that they are all ...

**4**

votes

**0**answers

104 views

### An inequality from the “Interlacing-1” paper

This question is in reference to this paper, http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf (or its arxiv version, http://arxiv.org/abs/1304.4132)
For the argument to ...

**3**

votes

**0**answers

35 views

### About the partial expectation polynomials in the Interlacing-I paper and perfect matchings

I am thinking of the polynomials $f_{s_1,s_2,..,s_k}$ as in the definition 4.3 in this paper http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf
In the use of these ...

**4**

votes

**0**answers

63 views

### Determinant of quotient of unbounded operators

I have been trying to prove this for a while but failed so far.
Let $A$ and $B$ are two positive, self-adjoint operators with compact resolvent on a Hilbert space $H$ defined on the same dense ...

**0**

votes

**0**answers

64 views

### $l_{\infty}$ norms of matrix perturbations

Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension.
What needs to be the bounds on (which?) norm of $B$ to ensure that $\lambda_{max}...

**5**

votes

**0**answers

227 views

### The Spectrum of certain differential operators

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$.
We consider the following polynomial vector field on ...

**2**

votes

**0**answers

340 views

### Largest eigenvalues distribution of tridiagonal symmetric random matrix

I would like to find the largest eigenvalue distribution of the following tridiagonal symmetric random matrix in an analytic way.
All the ${\lambda}_i$ are distributed the same way with chi-square (...

**2**

votes

**0**answers

43 views

### Error bounds for eigenvalue expansion of the Mathieu equation

The Mathieu equation is an important eigenvalue problem in Mathematical Physics that is completely understood in its properties, although there is no "direct way" of expressing eigenvalues and ...

**0**

votes

**1**answer

162 views

### Decomposition of a regular graph and connected subgraphs

I have asked almost same question earlier. I have been told that my question was poorly written, so I am trying to write it more clearly in this post. Also, this time I would be a little different in ...

**3**

votes

**3**answers

193 views

### Limits for eigenvalues for the Dirichlet Laplacian

If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem
$$
\begin{cases}
-\Delta u=\lambda u & \mbox{in }\Omega\\
u=0 & \mbox{on }\partial\...

**1**

vote

**0**answers

376 views

### Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition

Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected.
The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...

**1**

vote

**0**answers

69 views

### Interpreting (Fiedler) spectral bisectioning

I would appreciate help on how to interpret the results of spectral bisectioning of a graph.
Given a $G=(V,E)$ with size $N$ represented by $Q$ its Laplacian matrix where the eigenvalues are ordered ...

**0**

votes

**0**answers

45 views

### Column Inner Products vs. Row Inner Products

Given two matrices $A,B\in\mathbb{R}^{n\times r}$ where $A$ has orthogonal columns and $A^TB$ is symmetric, are there any non-trivial interesting relationships / inequalities between the following ...

**2**

votes

**1**answer

49 views

### Information on special matrices similar to Jacobi matrices

Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. One can arrive at these operators while studying discrete systems of particles interacting with ...

**10**

votes

**2**answers

370 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:
(...

**1**

vote

**1**answer

143 views

### Ask the validity of Tauberian lemma in Sogge's book

In C.D.Sogge's Fourier Integrals in Classical Analysis pp.128-129, he proved Lemma4.2.3(Tauberian Lemma):
Lemma. Let$g(\lambda)$ be a piece-wise continuous tempered function of $\mathbb{R}$. Assume ...

**2**

votes

**1**answer

294 views

### Is there any way to compare between diagonals of a resolvent and a Cauchy transform?

Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = ...

**3**

votes

**1**answer

90 views

### Bounds for smallest eigenvalue of Schrödinger equation with a superpotential and periodic boundary conditions

A very specific question, but posted on the off-chance that someone may be able to help. If we have a Schrödinger equation with arbitrary "superpotential"
\begin{equation}
-\frac{d^2 \psi}{dx^2} ...

**2**

votes

**2**answers

359 views

### Can one estimate the distribution of eigenvalues of a matrix by its Cauchy/Stieltje transform?

Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$
Is there any information ...

**1**

vote

**0**answers

87 views

### About properties of polynomials with common interlacing

Say $\{a_1,a_2,..,a_n\}$ and $\{b_1,b_2,...,b_n\}$ be the real roots of two monic polynomials of degree $n$ which have a common interlacing. (say I have arranged the roots in increasing order)
Can ...

**2**

votes

**1**answer

95 views

### References about spectral theory on Hyperbolic space

Can anyone suggest me some books or papers that include details about spectral theory on Hyperbolic spaces or related topics such as scattering theory on Hyperbolic spaces?
After some googling, I ...

**8**

votes

**2**answers

402 views

### Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold

It has been proved by Li-Yau and Zhong-Yang that if $M$ is a closed Riemannian manifold of dimension $n$ with nonnegative Ricci curvature, then the first nonzero eigenvalue $\lambda_1(M)$ of the (...

**0**

votes

**1**answer

139 views

### When can two Cauchy transforms intersect?

Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = \frac{q'...