Questions tagged [spectral-gap]

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Discrete spectrum of $A \otimes 1+ 1 \otimes B$ [migrated]

Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$, with both non-empty discrete spectra. Let us say, for instance, $\inf \, \sigma(A) = \lambda_1^A$ and $...
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Spectral gap of a Markov operator on $L^2$ with a symmetric $L^\infty$ kernel

Let $I$ be a compact interval, say $I:=(0,1)$, and $k\in L^\infty(I\times I)$ a symmetric Markov kernel, i.e. $k(x,y)=k(y,x)$ and $$\int_I k(x,y) d y = 1\qquad\mbox{for almost all } x\in I.$$ Let $K:L^...
Johny B's user avatar
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2 votes
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Spectral gaps and convergence of solutions

I'm reading the following paper by C. Villani on the Fokker-Planck equation $$u_t = Lu$$ with $L = \Delta - \nabla V\cdot \nabla$ in which he states (pg. 3) that the existence of a spectral gap for ...
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Flat function with a spectral gap

I am looking for a sequence of functions $f_n,n\geq 1$ in $L^2(\mathbb R)$ such that $f_n$ is equal to $1$ on $[-n,n]$ and $\hat{f_n}$ vanishes on $[-1,1]$. Actually, I would also like $f_n$ to be $...
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3 votes
1 answer
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Mixing time and spectral gap for a special stochastic matrix

Consider the following dimension stochastic matrix, \begin{bmatrix} p & q & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 &...
Hao Yuan's user avatar
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Lower bound on Spectral Gap of Rank one + Diagonal

For some $x\in\mathbb{R}^n, \|x\|_2^2=1$ and $\alpha\geq 0$, consider the positive semi-definite matrix $$ X_\alpha := xx^T + \alpha\sum_{k=1}^nx_k^2e_ke_k^T. $$ Suppose for simplicity that the ...
squattyroo's user avatar
3 votes
2 answers
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Estimating a spectral gap

Suppose you have a real positive definite matrix $A$ who eigenvalues are $\lambda_{1} \leq \lambda_{2} \leq \ldots \leq \lambda_{n}$. I am interested in bounding from below $\lambda_{2}-\lambda_{1}$. (...
Felix Goldberg's user avatar
20 votes
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Could unramified Galois groups satisfy a version of property tau?

This is an experiment: there is a question I want to mention in an article I'm writing, and I am not sure it's a sensible question, so I will ask it here first, in the hopes that if it's insensible ...
JSE's user avatar
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