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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

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        votes
        0answers
        28 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 ...
        2
        votes
        0answers
        61 views

        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 ...
        2
        votes
        0answers
        61 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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        votes
        0answers
        56 views

        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 $n\in\mathbb N$ $(\...
        1
        vote
        0answers
        44 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 ...
        10
        votes
        2answers
        451 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 ...
        3
        votes
        1answer
        53 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 $0$. ...
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        vote
        0answers
        47 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 ...
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        votes
        0answers
        46 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\...
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        votes
        2answers
        143 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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        vote
        0answers
        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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        vote
        0answers
        41 views

        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 ...
        6
        votes
        1answer
        136 views

        Weyl law for (non-semiclassical) Schrodinger operator

        The Weyl law for a semiclassical Schrodinger operator $$ A_h\ := \ -h^2\Delta+V(x) $$ on an $d$-dimensional complete Riemannian manifold $M$ says that the number $N(A_h,1)$ of eigenvalues of $A_h$ ...
        3
        votes
        1answer
        266 views

        Oscillatory integrals

        Consider the integrals $$I_n(\zeta,\epsilon)=\int_{-\zeta}^\zeta \left|(t-i\epsilon)^{-n}-(t+i\epsilon)^{-n}\right|\,dt$$ I would like to know the asymptotic behavior of $I_n(\zeta,\epsilon)$ for ...
        2
        votes
        0answers
        56 views

        Smallest singular value distribution

        Let $G_\mathbb{R}\in\mathbb{R}^{n\times n}$ and $G_\mathbb{C}\in\mathbb{C}^{n\times n}$ denote the real and complex Ginibre random matrices, i.e. random matrices with independent real/complex Gaussian ...

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