# 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

**1**

vote

**0**answers

106 views

### How much classical deformation theory do spectral sequences give you for free?

Many foundational results in deformation theory are proven using long, tedious and unsatisfying diagram chases, of the sort that I suspect could be `automated' using spectral sequences. For instance,
...

**0**

votes

**2**answers

111 views

### Spectrum equals eigenvalues for unbounded operator

Let $D$ be an unbounded densely defined operator on a separable Hilbert space $H$. If $D$ is diagonalisable with all eigenvalues having finite multiplicity and growing towards infinity, does it follow ...

**0**

votes

**0**answers

43 views

### Resolvent estimate of compact perturbation of self-adjoint operator

Let $T$ be a selfadjoint operator on Hilbert space $H$. Then we know that there is a resolvent estimation $$\left\lVert (\lambda-T)^{-1}\right\rVert = \frac{1}{dist(\lambda,\sigma(T))}, \ \lambda \in \...

**3**

votes

**1**answer

118 views

### Lower estimate of the minimal eigenvalue of a Hamiltonian

Consider a linear operator $H\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$ given by
$$H(\psi)(x):=-\Delta\psi(x)+V(x)\cdot \psi(x),$$
where $V\colon \mathbb{R}^3\to \mathbb{R}$ is a continuous (or ...

**2**

votes

**0**answers

91 views

### Limit circle/point of an ODE with finite eigenvalues

Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...

**5**

votes

**1**answer

81 views

### Connection between rates of convergence in ergodic theorems and spectral gap property

I've been reading Quantitative ergodic theorems and their number-theoretic applications By Gorodnik and Nevo (arXiv:1304.6847). Early on, there is a comment on rates of convergence in the mean ergodic ...

**5**

votes

**0**answers

350 views

### Spectral theory without topology

How much of spectral theory can be developed just working with vector spaces (finite or infinite dimensional) without referring to a choice of topology ?
Something along these lines, for example: ...

**5**

votes

**0**answers

66 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

**3**answers

97 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

145 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

124 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

104 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**

votes

**0**answers

49 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

215 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{...

**1**

vote

**0**answers

72 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 ...