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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

        3
        votes
        1answer
        90 views

        Mixing time and spectral gap for a special stochastic matrix

        Conisder 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 &...
        4
        votes
        1answer
        130 views

        The exceptional eigenvalues and Weyl's law in level aspect

        The Weyl law for Maass cusp forms for $SL_2(\mathbb Z)$ was obtained by Selberg firstly and has been generalized to various caces (Duistermaat-Guillemin, Lapid-Müller and others). For example, one of ...
        4
        votes
        0answers
        57 views

        Perturbation theory compact operator

        Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$ $\Vert Kx-\lambda x \Vert \le \varepsilon.$ It is well-known ...
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        votes
        0answers
        63 views

        Off-diagonal estimates for Poisson kernels on manifolds

        Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...
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        vote
        0answers
        162 views

        One question about Schrodinger Semigroups-(B. Simon)

        This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3). On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...
        2
        votes
        1answer
        319 views

        Compact operators on Banach spaces and their spectra

        I have a question about compact operators on Banach spaces. Let $B$ be a real Banach space and $L$ a closed linear operator on $B$. We assume that $L$ generates a contraction semigroup $\{T_t\}_{t>...
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        votes
        0answers
        116 views

        A special sequence

        I m looking for a sequence $(f_j)\in C^\infty(\Bbb{R})$ such that $$ \int^\infty_0\Big|\partial^2_r f_j+\frac{1}{r}\partial_r f_j+r^2f_j\Big|^2rdr\to 0, $$ and $$\int_{\Bbb{R^+}}|f_j(r)|^2 rdr=1\...
        2
        votes
        1answer
        89 views

        Non-isolated ground state of a Schrödinger operator

        Question. Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schr?dinger operator $-\...
        6
        votes
        1answer
        152 views

        Eigenvalue estimates for operator perturbations

        I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading. What was behind ...
        5
        votes
        1answer
        163 views

        Perturbing a normal matrix

        Let $N$ be a normal matrix. Now I consider a perturbation of the matrix by another matrix $A.$ The perturbed matrix shall be called $M=N+A.$ Now assume there is a normalized vector $u$ such that $\...
        4
        votes
        2answers
        79 views

        Stable matrices and their spectra

        I am a graduate student in engineering and we work a lot with so-called Hurwitz (or stable) matrices. A matrix in our terminology is called stable if the real part of the eigenvalues is strictly ...
        3
        votes
        1answer
        131 views

        Real part of eigenvalues and Laplacian

        I am working on imaging and I am a bit puzzled by the behaviour of this matrix: $$A:=\left( \begin{array}{cccccc} 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 &...
        3
        votes
        1answer
        128 views

        Spectrum of this block matrix

        Consider the following block matrix $$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix}?\right)$$ where all submatrices are square and matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...
        1
        vote
        1answer
        161 views

        Graph Laplacian Operator

        Consider the linear operator $\mathbb{L} : L^2([0,1])\to L^2([0,1])$ defined by $$ (\mathbb{L}f)(x) = \int_0^1 xy(f(x)-f(y)) \mathrm{d}y $$ for all $f\in L^2([0,1])$ and $x \in [0,1]$. Is $\mathbb{L}$...
        1
        vote
        2answers
        109 views

        Self-adjoint extensions for pseudo-differential operators

        The class $\Sigma^1$ of symbols on $\mathbb R^{2n}$ is made with $C^\infty$ functions $a$ of $X=(x,\xi)\in \mathbb R^n\times\mathbb R^n$ such that $$ \vert\partial_X^\alpha a\vert\le C_\alpha(1+\vert ...

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