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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

630 questions
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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 ...
473 views

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 ...
55 views

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 $0$. ...
48 views

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 ...
47 views

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\...
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 ...
35 views

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 ...

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