**1**

vote

**0**answers

111 views

### On isolated points of the approximate point spectrum of a bounded operator

Let $X$ be a complex Banach space and $T$ be a bounded operator acting on $X$.
Let $\sigma(T)$ and $\sigma_{ap}(T)$ denote the spectrum and approximate point spectrum of $T$, respectively.
Let ...

**0**

votes

**0**answers

64 views

### Spectral theory of $\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2}$ and $4 \frac{\partial ^2}{\partial z\partial \bar{z}}$ [on hold]

Since $\mathbb R^2 \simeq \mathbb C$ and $\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2} =4 \frac{\partial ^2}{\partial z\partial \bar{z}}$, I like to know is that the spectral ...

**1**

vote

**1**answer

58 views

### Spectral radius's relation with row sum

Let $A$ be a non-negative $N \times N$ square matrix with $a_{i,i}=0, 1 \leq i \leq N$. Also, let $r_i$ be the $i$-th row sum of $A$.
I know that $\rho(A)$, the spectral radius of $A$, is bounded as ...

**-6**

votes

**1**answer

54 views

### Does $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ holds for matrices with spectral radius smaller then 1?

Given a symmetric positive semidefinite matrix matrix $A$, if its spectral radius $0<\rho(A)<1$, does the inequality $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ hold true?
$\|A\|_{2}$ denotes ...

**1**

vote

**1**answer

75 views

### Largest element in inverse of a positive definite symmetric matrix [closed]

If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...

**0**

votes

**0**answers

55 views

### What equals $\ker[(A-\lambda I)^+]$ for a negative unbounded operator $A$?

We have the following result:
$\{ E_{\lambda}; \, -\infty <\lambda < + \infty\}$ is a spectral family, where $E_{\lambda}$ is the projection of $H$ onto the null space $\mathscr N ...

**2**

votes

**1**answer

253 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) = ...

**2**

votes

**0**answers

46 views

### When does the ground state energy continuously depend on a parameter?

Given a family of Schrödinger operators $H_\gamma=-\Delta+V_\gamma$, under which condition is the map $\gamma\mapsto\inf\sigma(H_\gamma)$ continuous?
This is surely the case for many textbook ...

**4**

votes

**1**answer

150 views

### Recovering Spherical Harmonics from Discrete Samples

Consider a collection of $N$ points on the 2-sphere chosen uniformly at random. Let's say that there's an edge between two such vertices if their geodesic distance is less than $r_N$. The resulting ...

**5**

votes

**1**answer

797 views

### Why is the cuspidal spectrum discrete?

I have a short question concerning the spectral theory of automorphic forms. What is the main property of the unipotent group $N$, which consist of matrices in the form \begin{pmatrix} 1 & t \\ 0 ...

**2**

votes

**0**answers

34 views

### Nonnegative Inverse Eigenvalue Problem (NIEP),

Does the NIEP, currently open for $n\ge 5$, have any good, practical applications?
For the easy case, $n=2$, I am able to prove some of the results that agree with the current literature.
In some of ...

**1**

vote

**1**answer

126 views

### Can we claim that all the terms in a matrix are less than equal to 1 if spectral radius is less than 1?

I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using,
H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term ...

**1**

vote

**0**answers

51 views

### inverse of asymptotic Toeplitz matrix with band limited associated function

I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation.
...

**3**

votes

**1**answer

92 views

### Restrictions on spectral measure

Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$
Here ...

**2**

votes

**0**answers

235 views

### A problem on Hecke operators

I have a problem that is related to page 11 in http://www.cims.nyu.edu/~venkatesh/research/ml.pdf
Fix $m\in\Bbb N$.
Pick $(a,b)\in\Bbb Z_p^2$ randomly such that $m<a,b<2m$ with $a,b$ ...

**52**

votes

**2**answers

810 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 ...

**0**

votes

**1**answer

90 views

### Spectrum of compact operator between different Banach spaces

Let $X, Y$ be two different Banach spaces, and let $T: X \to Y$ be a compact linear operator. Suppose the identity $I : X \to Y$ is well-defined. (For example, we could have $X = L^2([0,1])$ and $Y = ...

**6**

votes

**3**answers

220 views

### Non-self adjoint Sturm-Liouville problem

I'm new to this site, but I felt the need to post when I recently came in to an ordinary differential equation/boundary value problem with this form:
$(1)- \frac{d^2 y}{d x^2} + \frac{m(m+1)} ...

**1**

vote

**0**answers

59 views

### inverse problem to resolution of the identity

Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define ...

**20**

votes

**2**answers

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 ...

**0**

votes

**1**answer

126 views

### Continuity of the largest eigenvalue with respect to length

Let $k:\mathbb{R}^+\to\mathbb R^+$ be a continuous function. For $a>0$, define $T_a$ acting on $L^2[0,a]$, by
$$T_af(x) = \int_0^a k(|x-y|)f(y)\,dy.$$
Clearly for each $a>0$, the operator ...

**1**

vote

**1**answer

120 views

### Zero set of eigenfunction along a sub manifold

Let $M$ be a 2-dimensional closed Riemannian manifold and let $$\phi:M\rightarrow M$$ be an isometry with $\phi^2=Id_M$. Consider the fixed point set $$F:=\lbrace x\in M: \phi(x)=x \rbrace\subset M,$$ ...

**7**

votes

**0**answers

102 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 ...

**10**

votes

**2**answers

204 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 ...

**4**

votes

**1**answer

175 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 ...

**2**

votes

**1**answer

120 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 ...

**1**

vote

**0**answers

89 views

### Eigenvalues of the D'Alembertian operator

My question about the spectral theory of the D'Alembertian operator on a Lorentzian manifolds (say the spacetime $M^{3+1}$) given by $$\square = -\partial_{t}^2 + \Delta$$ for the metric $g=(-+++)$. ...

**4**

votes

**1**answer

96 views

### For self-adjoint $T$ on $L^2(\mathbb{R}^n)$, when does $(1 + |x|)^{-1} (T - i \varepsilon)^{-1}(1 + |x|)^{-1}$ have a limit as $\varepsilon \to 0$?

Let $T : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ be a possibly unbounded self adjoint operator. Let $R_\varepsilon$ denote the resolvent $(T - i \varepsilon)^{-1}$, $\varepsilon > 0$. Suppose ...

**2**

votes

**1**answer

65 views

### Integrating the resolvent of a self-adjoint operator across a continuous part of the spectrum

Let $A$ be a closed self-adjoint operator on a Hilbert space $H$, possibly unbounded and hence defined on a dense domain $D(A) \subset H$. It is well known that integrating the resolvent $R_z = (z I - ...

**0**

votes

**1**answer

108 views

### Uniform continuity of spectrum as function of operator [closed]

It is well known that the spectrum is continuous as function of operator. More precisely, let $\mathcal{H}$ be separable Hilbert space and $\mathcal{B}(\mathcal{H})$ the Banach algebra of linear ...

**4**

votes

**0**answers

53 views

### Lower bound on the sum of singular values for a sum of Hermitian matrices

Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that ...

**7**

votes

**1**answer

140 views

### Spectral mapping theorem for polynomials in $z,\overline{z}$ and direct construction of the function calculus for a normal operator

Suppose that $A$ is an element in Banach algebra and $p$ is a polynomial. Then we have an equality $p(\sigma(A))=\sigma(p(A))$ where $p(A)$ has an elementary meaning. This theorem (the spectral ...

**8**

votes

**1**answer

124 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 ...

**4**

votes

**2**answers

156 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 ...

**1**

vote

**0**answers

60 views

### Partial trace with spectral measure

I'm a physicist who needs mathematical advice:
Let $A= \sum_{i=1}^{\infty} a_i P_{\phi_i}$ be a self-adjoint operator with projectors $P_{\phi_i}$ on the orthonormal eigenbasis $(\phi_i).$ Let $$B= ...

**11**

votes

**1**answer

674 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 ...

**3**

votes

**0**answers

243 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 ...

**11**

votes

**6**answers

2k 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 ...

**4**

votes

**3**answers

174 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

47 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:
...

**0**

votes

**0**answers

79 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

85 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

38 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
...

**3**

votes

**0**answers

89 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*}
...

**3**

votes

**1**answer

101 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. ...

**6**

votes

**0**answers

143 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 ...

**33**

votes

**10**answers

8k 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?

**3**

votes

**0**answers

69 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 ...

**9**

votes

**1**answer

351 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) ...