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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

        1
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        0answers
        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
        2answers
        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 ...
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        0answers
        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
        1answer
        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
        0answers
        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
        1answer
        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
        0answers
        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: ...
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        votes
        0answers
        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
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        3answers
        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
        1answer
        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 ...
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        votes
        0answers
        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 $...
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        votes
        0answers
        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
        0answers
        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
        1answer
        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{...
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        vote
        0answers
        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 ...

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