Search results for "Mathematics::Spectral Theory"
showing 10 items of 111 documents
Two non-zero solutions for Sturm–Liouville equations with mixed boundary conditions
2019
Abstract In this paper, we establish the existence of two non-zero solutions for a mixed boundary value problem with the Sturm–Liouville equation. The approach is based on a recent two critical point theorem.
Estimates for Sums of Eigenvalues of the Free Plate via the Fourier Transform
2017
Using the Fourier transform, we obtain upper bounds for sums of eigenvalues of the free plate.
Clarkson-McCarthy inequalities with unitary and isometry orbits
2020
Abstract A refinement of a trace inequality of McCarthy establishing the uniform convexity of the Schatten p-classes for p > 2 is proved: if A , B are two n-by-n matrices, then there exists some pair of n-by-n unitary matrices U , V such that U | A + B 2 | p U ⁎ + V | A − B 2 | p V ⁎ ≤ | A | p + | B | p 2 . A similar statement holds for compact Hilbert space operators. Another improvement of McCarthy's inequality is given via the new operator parallelogramm law, | A + B | 2 ⊕ | A − B | 2 = U 0 ( | A | 2 + | B | 2 ) U 0 ⁎ + V 0 ( | A | 2 + | B | 2 ) V 0 ⁎ for some pair of 2n-by-n isometry matrices U 0 , V 0 .
Singular Neumann (p, q)-equations
2019
We consider a nonlinear parametric Neumann problem driven by the sum of a p-Laplacian and of a q-Laplacian and exhibiting in the reaction the competing effects of a singular term and of a resonant term. Using variational methods together with suitable truncation and comparison techniques, we show that for small values of the parameter the problem has at least two positive smooth solutions.
Weyl's theorem for perturbations of paranormal operators
2007
A bounded linear operator T ∈ L(X) on a Banach space X is said to satisfy "Weyl's theorem" if the complement in the spectrum of the Weyl spectrum is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this paper we show that if T is a paranormal operator on a Hilbert space, then T + K satisfies Weyl's theorem for every algebraic operator K which commutes with T.
Ultrarelativistic bound states in the spherical well
2016
We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator $(-\Delta )^{1/2}$, whose action is restricted to functions that vanish beyond the interior of a unit sphere in three spatial dimensions. We provide high accuracy spectral datafor lowest eigenvalues and eigenfunctions of this infinite spherical well problem. Our focus is on radial and orbital shapes of eigenfunctions. The spectrum consists of an ordered set of strictly positive eigenvalues which naturally splits into non-overlapping, orbitally labelled $E_{(k,l)}$ series. For each orbital label $l=0,1,2,...$ the label $k =1,2,...$ enumerates consecutive $l$-th series eigenvalues. Each of them is $2l+1$-degenerate. …
Pseudodifferential operators of Beurling type and the wave front set
2008
AbstractWe investigate the action of pseudodifferential operators of Beurling type on the wave front sets. More precisely, we show that these operators are microlocal, that is, preserve or reduce wave front sets. Some consequences on micro-hypoellipticity are derived.
Tensor products, multiplications and Weyl’s theorem
2005
Tensor productsZ=T 1⊗T 2 and multiplicationsZ=L T 1 R T 2 do not inherit Weyl’s theorem from Weyl’s theorem forT 1 andT 2. Also, Weyl’s theorem does not transfer fromZ toZ*. We prove that ifT i,i=1, 2, has SVEP (=the single-valued extension property) at points in the complement of the Weyl spectrumσ w(Ti) ofT i, and if the operatorsT i are Kato type at the isolated points ofσ(Ti), thenZ andZ* satisfy Weyl’s theorem.
Interior Eigenvalue Density of Jordan Matrices with Random Perturbations
2017
International audience; We study the eigenvalue distribution of a large Jordan block subject to a small random Gaussian perturbation. A result by E. B. Davies and M. Hager shows that as the dimension of the matrix gets large, with probability close to 1, most of the eigenvalues are close to a circle.We study the expected eigenvalue density of the perturbed Jordan block in the interior of that circle and give a precise asymptotic description.; Nous étudions la distribution de valeurs propres d’un grand bloc de Jordan soumis à une petite perturbation gaussienne aléatoire. Un résultat de E. B. Davies et M. Hager montre que quand la dimension de la matrice devient grande, alors avec probabilité…
Note on singular Clairaut-Liouville metrics
2008
Computations on Clairaut-Liouville metrics on S^2 with a finite order singularity.