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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

      2
      votes
      0answers
      27 views

      Existence of a fixed point for this operator

      I'm looking for some mathematical results that I might be able to apply to see if an operator I'm considering has a fixed point. In particular consider, $$ Ag(x) = \Big\{ \xi(x) + K(g(x))^\frac{1}{\...
      1
      vote
      0answers
      34 views

      The Morse Index of a $T$- periodic geodesics is a integer number?

      It is well known that compact Riemannian manifolds $(M, g)$ with periodic geodesic flows have ( Besse Book) exceptional spectral properties: the spectrum of $ \sqrt{ - \Delta}$, the square root of ...
      3
      votes
      1answer
      121 views

      Quasinilpotent vectors of Newton potential vanish

      Suppose $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$. Consider the Newton potential \begin{equation} T [\phi](x) = \int_{\Omega} \frac{1}{|x-y|} \phi(y)dy. \end{equation} It is well know ...
      3
      votes
      1answer
      93 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
      134 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
      58 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 ...
      2
      votes
      0answers
      64 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
      321 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>...
      0
      votes
      0answers
      118 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
      132 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
      158 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
      81 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
      132 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 &...

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