# 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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### Sum of Square of the Eigenvalues of Wishart Matrix

Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...

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### Shift operator on a Banach space

I have been the paper titled Dual Piecewise Analytic Bundle Shift Models of Linear Operators by Dmitry Yakubovich.
In the second paragraph of the introduction it says "Let $T$ be a bounded Linear ...

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146 views

### Graph Fourier transform definition

I have a question about the definition of the graph Fourier transform. Let me start with definition.
Let $A$ be the adjacency matrix of a graph $G$ with vertex set $V = \{1, 2, \dots, n\}$. The ...

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35 views

### Zero in the spectrum of an elliptic second order operator

This might be considered as a continuation of my previous question Spectrum of a linear elliptic operator
but is independent. I have another question on V. Gribov's paper "Quantization of non-Abelian ...

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### Spectrum of a linear elliptic operator

In the paper in quantum fields theory by
Gribov,V.; (1978) "Quantization of non-Abelian gauge theories". Nuclear Physics B. 139: 1–19;
in Section 3 the author makes the following claim from PDE and ...

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65 views

### Spectrum of a Hamiltonian which is a perturbation of Laplacian

Let $\Delta =\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}$ be the Laplacian on $\mathbb{R}^3$.
Consider a self adjoint operator $H$ on complex ...

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### Show convergence of a sequence of resolvent operators

Let
$E$ be a locally compact separable metric space
$(\mathcal D(A),A)$ be the generator of a strongly continuous contraction semigroup on $C_0(E)$
$E_n$ be a metric space for $n\in\mathbb N$
$(\...

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47 views

### Strong Differentiability of Spectral Projections

Let $H$ be a Hilbert space and $W$ be a dense subspace, equipped with a different norm that turns it into a Hilbert space. Let $(A(t))_{t\in[0,T]}$ be a family of Operators in $B(W,H)$ (bounded ...

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### Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifold

Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta_g$ be its Laplace-Beltrami operator. A "well-known fact" is that the eigenvalues of $\Delta_g$ have finite multiplicity and tend to ...

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### How does $E$ closed follow from the upper semicontinuity of the spectrum?

Let $f$ be an analytic function for a domain $D$ of $\mathbb{C}$ into a Banach algebra $A$. Suppose that, for all $\lambda \in D$, $\text{Sp}f(\lambda)$ is finite or a sequence converging to $0$.
...

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### Bounds on spectral radius using chromatic number

I am struggling with this question:
If I have a connected graph $G$ on $n$ vertices and $m$ edges with chromatic number $d$ then how can I give a bound(lower and upper) on its spectral radius in ...

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### Construction of a special sequence [duplicate]

I m looking for a sequence $(f_j)\in C^\infty(\Bbb{R})$ such that
$$
\int^\infty_0\Big|4r\partial^2_r f_j(r)+4\partial_r f_j(r)+rf_j(r)\Big|^2dr\to 0,
$$ and
$$\int_{\Bbb{R^+}}|f_j(r)|^2 dr=1\quad\...

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144 views

### Invariance of spectrum under conjugation

Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous ...

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### Stable region of minimal hypersurfaces with finite Morse index

In this Inventiones Mathematicae paper, Fischer-Colbrie proved the following result (Proposition 1):
Proposition: Let $ M$ be a complete two-sided minimal surface in a three manifold $N$. Then if $M$...

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### Sturm-Liouville-like Eigenproblem

Consider the piecewise-deterministic Markov process on $\mathbf{R}$ which
moves according to the vector field $\phi (x) = 1$,
experiences events at rate $\lambda(x) = 1$, and
at events, jumps ...