# Questions tagged [sp.spectral-theory]

Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices

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55 views

### Laplace Beltrami eigenvalues on surface of polytopes

The recently posted arxiv paper Spectrum of the Laplacian on Regular Polyhedra
by Evan Greif, Daniel Kaplan, Robert S. Strichartz, and Samuel C. Wiese, collects numerical evidence for conjectured ...

**0**

votes

**0**answers

73 views

### How to understand the sign periodicity of any conditionally convergent series? [closed]

Given any conditionally convergent series $\sum_{n\geq1} a_n$. I wonder if there is a "standard way/method" to "investigate/estimate" the sign periodicity of $a_n$ by some explicit functions $f(x)$ i....

**0**

votes

**3**answers

93 views

### Clustering on tree

I am looking for a method that would identify clusters in a tree-like structure. In the figure below you can see a very simple example where one can visually identify distinct branches with a lot of ...

**8**

votes

**1**answer

138 views

### Spectrum of a first-order elliptic differential operator

Suppose that I have a first-order elliptic differential operator $A: \mathrm{dom}(A) \subset L^2(E) \to L^2(E)$, where $(E,h^E) \to M$ is a hermitian vector bundle and $M$ is a compact manifold.
I ...

**0**

votes

**0**answers

119 views

### Sudden appearance of an eigenvalue of a self-adjoint operator $H = H_0 + \lambda H_1$

In doing some numerical calculation in quantum mechanics, we found something surprising to us. Let the Hamiltonian be
$$ H = H_0 + \lambda H_1 , $$
where both $H_0$ and $H_1$ are self-adjoint, and $...

**2**

votes

**0**answers

96 views

### Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum

I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...

**2**

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43 views

### First eigenvalue of the spherical cap

Let $S$ be the round $n$-sphere of radius $R$ in Euclidean space, and let $r$ be the intrinsic distance from the north pole. Further, let $U(r)$ be the spherical cap of intrinsic radius r. (So $U(0)$ ...

**3**

votes

**1**answer

211 views

### Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator

For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...

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vote

**0**answers

70 views

### asymptotic behaviour of principal eigenfunctions and Large Deviations

Dear Math Overflowers,
I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...

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**0**answers

47 views

### Theta Summable operator with bounded trace

Let $D$ be an unboudned self-adjoint operator on the Hilbert space $H$. We assume that all spectrum of $D$ are eigenvalues and $D$ is theta-summable, i.e. $e^{-tD^2}$ is of trace class for all $t>...

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votes

**1**answer

120 views

### The effect of random projections on matrices

Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$.
Suppose $\frac{A+...

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votes

**0**answers

167 views

### Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?

Let $X$ be a smooth, Riemannian manifold. It is known that the geometry of $X$ can be recovered from its heat kernel $k_{t}(x,y)$, using Varadhan's Lemma: $\displaystyle\lim_{t \to 0} t \log k_{t}(x,y)...

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vote

**1**answer

52 views

### Efficient way to compute eigenvalue decomposition for following problem

I have an optimization problem
$$\begin{array}{ll} \text{minimize} & Tr(X^TAX) \\ \text{subject to} & X^TX=I
\end{array}$$
where $A\in R^{n \times n}$ and it is symmetric positive definite, ...

**0**

votes

**0**answers

71 views

### Bounded and sectorial operators

Is there any assumption for a bounded operator to be sectorial ? Is there any characterization of such operators ?
Here, the definition of sectorial operators follows the book of Markus Haase:
...

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votes

**2**answers

242 views

### Matrix rescaling increases lowest eigenvalue?

Consider the set $\mathbf{N}:=\left\{1,2,....,N \right\}$ and let $$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\...

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50 views

### Characterizing the separability of the Gelfand space of a semisimple commutative Banach algebra

Problem. Is the separability of the Gelfand space of a semi-simple commutative Banach algebra $A$ equivalent to the existence of a countable family $\{\varphi_n\}_{n\in\omega}$ of multiplicative ...

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votes

**2**answers

595 views

### Eigenvalues of Laplace-Beltrami on half sphere

Let $ \Delta_\theta$ denote the Laplace-Beltrami operator on $S^{N-1}$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^{N-...

**4**

votes

**1**answer

171 views

### Intuitive proof of Golden-Thompson inequality

Sutter et al. [1] in their paper "Multivariate Trace Inequalities" give an intuitive proof of the following Golden-Thompson inequality:
For any hermitian matrices $A,B$:
$$
\text{tr}(\exp{(A+B)}) \...

**-2**

votes

**1**answer

133 views

### sum of positive definite matrix

sum of positive definite matrix $A+B $is positive definite. I want to look at the spectrum of $C=A+B$
can we say the ith largest eigenvalue of $C$ is no less than the ith largest eigenvalue of $A$ i....

**5**

votes

**0**answers

52 views

### Eigenvalue lower bounds for manifold with positive Ricci curvature

For closed $n$-manifold with Ricci curvature $\ge (n-1)$, it is known that the first eigenvalue $\lambda_1\ge n$ with equality holds if and only if $M$ is isometric to the Euclidean sphere $S^n$. My ...

**4**

votes

**0**answers

261 views

### Spectral Gap of Elliptic Operator

Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled?
The boundary condition is that the ...

**2**

votes

**3**answers

274 views

### Non-linear Basis for PDE's

Asked this on stack exchange and got no response, so I'll try here.
An idea popped into my head awhile ago while doing a project on non-linear effects of systems of coupled oscillators, but I'm not ...

**4**

votes

**1**answer

84 views

### Non-point spectrum for diagonalisable self-adjoint unbounded operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...

**5**

votes

**1**answer

130 views

### Hilbert representation of a bilinear form

Let $\sigma:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ be a bilinear symmetric form which is non-degenerate in the sense that for every $0\neq u\in \mathbb{R}^n$ there is $v\in \mathbb{R}^n$ with $...

**4**

votes

**0**answers

109 views

### Spectral gap for the Brownian motion with drift on a compact manifold

Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...

**4**

votes

**2**answers

114 views

### Radial symmetry of the first eigenfunction

Let $M$ be a simply connected space form (i.e. $\mathbb R^n$, sphere, or hyperbolic space) and $B$ be a ball in $M$. Let $\phi$ be the first Laplacian eigenfunction on $B$, with respect to the ...

**1**

vote

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47 views

### Nodal domains on a surface

What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators?
In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \...

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vote

**0**answers

37 views

### First eigenvalue for domains in hyperbolic space

I am interested in examples of bounded open subsets of the hyperbolic space, for which the first eigenvalue of the Dirichlet Laplace operator (acting on functions) is known. In Euclidean space several ...

**4**

votes

**1**answer

185 views

### Can this self-adjoint operator have an infinite-dimensional compression with compact inverse?

The following might be quite straightforward, but I very rarely work in detail with unbounded operators, so I thought it would be worth seeing quickly if I have overlooked an example that is obvious ...

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votes

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181 views

### Absence of fixed points

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed ...

**2**

votes

**1**answer

84 views

### Local-Global Principle in Graph Spectrum

The question is a bit vague, but any ideas/directions will be appreciated.
Let us fix an $n$-vertex $d$-regular graph $G=(V,E)$. As I understand it, the eigenvalues of the adjacency matrix $A$ of $G$ ...

**16**

votes

**3**answers

517 views

### Does a spectral gap lift to covering spaces?

Let $M$ be a complete Riemannian manifold. Denote $\Delta_M\ge0$ the unique self-adjoint extension of the Laplace-Beltrami operator in $L^2(M)$ and $\sigma(\Delta_M)\subset [0,\infty)$ its spectrum. ...

**3**

votes

**0**answers

140 views

### Does the zeta regularized Laplacian determinant measure the volume of some parameter space? How many “spanning trees” on a manifold?

Let $(M,g)$ be a Riemannian manifold, with Laplacian $\Delta$. If $\lambda_i$ are the nonzero eigenvalues of $\Delta$, we can define the zeta function $\zeta(s) = \Sigma \lambda_i^{-s}$. By analytic ...

**4**

votes

**1**answer

250 views

### What is $e^{- \zeta_{\Delta} '(0)}$ for a $\Delta$ the Laplacian of a manifold?

For a connected, finite graph $G$, let $\lambda_1, \ldots, \lambda_n$ denote the nonzero eigenvalues of the graph Laplacian. We define $\zeta_G = \Sigma_{i = 1}^n \lambda_i^s$.
Then Kirkoffs Matrix-...

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votes

**2**answers

101 views

### Spectrum of finite-band random matrices?

Let
$X_n=(X_{ij})_{1 \leq i,j \leq n}$ such that :
$$ \begin{cases}
&X_{ij} = 0 \quad \text{if}\quad \vert i - j \vert > k\\
& X_{ij} \sim P_X \quad \text{otherwise}
\end{cases}$$
And ...

**0**

votes

**0**answers

55 views

### Numerical error on the spectrum of a matrix

Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small)...

**16**

votes

**1**answer

278 views

### Approximate eigenvectors for a set of non-commuting self-adjoint operators

This problem is motivated by finding the right mathematical setting for expressing the compatibility of classical physics with quantum mechanics.
Let $\mathcal H$ be a Hilbert space and $S$ a ...

**2**

votes

**2**answers

154 views

### iid random operator and its spectrum

consider an insteresting question:
given Banach Space $ \mathcal{B}$, independent identical distribution random operator on $ \mathcal{B}$: $ (T_i)_{i \ge 1} $, where operator space is endowed with ...

**3**

votes

**1**answer

57 views

### Simultaneous diagonalization on spaces with constant curvature

I have an operator $C$ that I wish to diagonalize on a Riemmanian manifold $M$ with constant curvature $\Lambda$
$$C = A + B$$
Now I know that these operators $A$ and $B$ commute in flat space, but on ...

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votes

**0**answers

79 views

### What is spectral multiplicity for multiplication operators in general von Neumann algebra set up?

When two multiplication operators $M_{f}$ and $M_{g}$ acting on $L^2(X,\mu) $and $L^2(Y,\nu)$ are unitary equivalent? How multiplicity function look like here? What is the spectral multiplicity in ...

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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 ...

**2**

votes

**1**answer

114 views

### A nodal theorem in 1D

Consider a 1D zero-energy Schrödinger equation on the half-line,
$(-\partial_x^2 + V(x))\psi(x)=0, \quad x \in (0, \infty)$
with a zero boundary condition $\psi(0) = 0$.
Is it true that if the zero-...

**5**

votes

**1**answer

141 views

### Spectrum of the product of operators

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $F$.
Let $A,B\in \mathcal{B}(F)^+:=\left\{T\in \mathcal{B}(F);\,\langle Tx, x\...

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71 views

### Discrete spectrum of Schrodinger operator

Assume $\Omega$ is a non-compact region or manifold with dimension $\geq4$. Let $H=-\Delta+V$ be Schrodinger operator. Here $V$ is a (smooth)function.
I know that if $V\geq c>0$ or $V\to c>0$,...

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55 views

### Density in the graph norm

Let $T\in B(H)$, where $H$ is a (separable) complex Hilbert space. Let $A$ be an unbounded self-adjoint operator acting in $H$. We denote by $D(A)$ its domain. We endow it with the graph norm, e.g., $\...

**1**

vote

**1**answer

69 views

### Moment generating function of spectral norm of iid N(0,1) data matrix

Let $W^{p\times p}$ be a normal data matrix with $W_{ij}$ i.i.d. $N(0,1)$. Are there any results on the evaluation, or upper bound for the Moment Generating Function of the spectral norm of W, that is,...

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21 views

### sensitive perturbation approximation

I was reading paper which associated with perturbation approximation. paper1 paper2.
In paper1:
$\bar{R}=R+\epsilon C$, first order: when $\Lambda_1\gg\Lambda_2$, $\Delta\Lambda_{max}=\frac{\vec{v}^...

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109 views

### Perturbation of Elliptic operator

Let $\Omega$ be an open region or a non-compact complete manifold, $L$ be an elliptic operator with possibly non vanishing zero-order term, e.g. $-\Delta+q$. Suppose $W$ is an operator such that $W(...

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39 views

### Bound of analytic torsion for a line bundle

Let $E\rightarrow X$ be a line bundle over a compact Riemann surface. Let $\Delta_{E}$ be the Dolbeault Laplacian $\Delta_{\overline{\partial}}$ extended to sections of $E$. Let $g_1, g_2$ be two ...

**4**

votes

**1**answer

95 views

### Reference for Weyl's law for higher order operators on closed Riemannian manifolds

I am looking at page 32 (beginning of Chapter 5) here. We are given a formally self-adjoint, metrically defined differential operator $A$ on $(M^n,g)$ of order $2l$ with positive definite leading ...