Search results for "Eigenvector"
showing 10 items of 303 documents
Regular solutions of transmission and interaction problems for wave equations
1989
Consider n bounded domains Ω ⊆ ℝ and elliptic formally symmetric differential operators A1 of second order on Ωi Choose any closed subspace V in , and extend (Ai)i=1,…,n by Friedrich's theorem to a self-adjoint operator A with D(A1/2) = V (interaction operator). We give asymptotic estimates for the eigenvalues of A and consider wave equations with interaction. With this concept, we solve a large class of problems including interface problems and transmission problems on ramified spaces.25,32 We also treat non-linear interaction, using a theorem of Minty29.
Gibbs states, algebraic dynamics and generalized Riesz systems
2020
In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some preliminary aspects of the Tomita-Takesaki theory in our context.
Intertwining operators for non-self-adjoint hamiltonians and bicoherent states
2016
This paper is devoted to the construction of what we will call {\em exactly solvable models}, i.e. of quantum mechanical systems described by an Hamiltonian $H$ whose eigenvalues and eigenvectors can be explicitly constructed out of some {\em minimal ingredients}. In particular, motivated by PT-quantum mechanics, we will not insist on any self-adjointness feature of the Hamiltonians considered in our construction. We also introduce the so-called bicoherent states, we analyze some of their properties and we show how they can be used for quantizing a system. Some examples, both in finite and in infinite-dimensional Hilbert spaces, are discussed.
Non-self-adjoint hamiltonians defined by Riesz bases
2014
We discuss some features of non-self-adjoint Hamiltonians with real discrete simple spectrum under the assumption that the eigenvectors form a Riesz basis of Hilbert space. Among other things, {we give conditions under which these Hamiltonians} can be factorized in terms of generalized lowering and raising operators.
Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents
1984
Abstract In this paper we study the existence of nontrivial solutions for the boundary value problem { − Δ u − λ u − u | u | 2 ⁎ − 2 = 0 in Ω u = 0 on ∂ Ω when Ω⊂Rn is a bounded domain, n ⩾ 3, 2 ⁎ = 2 n ( n − 2 ) is the critical exponent for the Sobolev embedding H 0 1 ( Ω ) ⊂ L p ( Ω ) , λ is a real parameter. We prove that there is bifurcation from any eigenvalue λj of − Δ and we give an estimate of the left neighbourhoods ] λ j ⁎ , λj] of λj, j∈N, in which the bifurcation branch can be extended. Moreover we prove that, if λ ∈ ] λ j ⁎ , λj[, the number of nontrivial solutions is at least twice the multiplicity of λj. The same kind of results holds also when Ω is a compact Riemannian manif…
Approximations of positive operators and continuity of the spectral radius III
1994
AbstractWe prove estimates on the speed of convergence of the ‘peripheral eigenvalues’ (and principal eigenvectors) of a sequence Tn of positive operators on a Banach lattice E to the peripheral eigenvalues of its limit operator T on E which is positive, irreducible and such that the spectral radius r(T) of T is a Riesz point of the spectrum of T (that is, a pole of the resolvent of T with a residuum of finite rank) under some conditions on the kind of approximation of Tn to T. These results sharpen results of convergence obtained by the authors in previous papers.
Toeplitz band matrices with small random perturbations
2021
We study the spectra of $N\times N$ Toeplitz band matrices perturbed by small complex Gaussian random matrices, in the regime $N\gg 1$. We prove a probabilistic Weyl law, which provides an precise asymptotic formula for the number of eigenvalues in certain domains, which may depend on $N$, with probability sub-exponentially (in $N$) close to $1$. We show that most eigenvalues of the perturbed Toeplitz matrix are at a distance of at most $\mathcal{O}(N^{-1+\varepsilon})$, for all $\varepsilon >0$, to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.
Perturbed eigenvalue problems for the Robin p-Laplacian plus an indefinite potential
2020
AbstractWe consider a parametric nonlinear Robin problem driven by the negativep-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation$$f(z,\cdot )$$f(z,·)is$$(p-1)$$(p-1)-sublinear and then the case where it is$$(p-1)$$(p-1)-superlinear but without satisfying the Ambrosetti–Rabinowitz condition. We establish existence and uniqueness or multiplicity of positive solutions for certain admissible range for the parameter$$\lambda \in {\mathbb {R}}$$λ∈Rwhich we specify exactly in terms of principal eigenvalue of the differential operator.
Multiple solutions for nonlinear nonhomogeneous resonant coercive problems
2018
We consider a nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a \begin{document}$p$\end{document} -Laplacian ( \begin{document}$2 ) and a Laplacian. The reaction term is a Caratheodory function \begin{document}$f(z,x)$\end{document} which is resonant with respect to the principal eigenvalue of ( \begin{document}$-\Delta_p,\, W^{1,p}_0(\Omega)$\end{document} ). Using variational methods combined with truncation and comparison techniques and Morse theory (critical groups) we prove the existence of three nontrivial smooth solutions all with sign information and under three different conditions concerning the behavior of \begin{document}$f(z,\cdot)$\end{document} near zero. By …
Eigenvalues of non-hermitian matrices: a dynamical and an iterative approach. Application to a truncated Swanson model
2020
We propose two different strategies to find eigenvalues and eigenvectors of a given, not necessarily Hermitian, matrix (Formula presented.). Our methods apply also to the case of complex eigenvalues, making the strategies interesting for applications to physics and to pseudo-Hermitian quantum mechanics in particular. We first consider a dynamical approach, based on a pair of ordinary differential equations defined in terms of the matrix (Formula presented.) and of its adjoint (Formula presented.). Then, we consider an extension of the so-called power method, for which we prove a fixed point theorem for (Formula presented.) useful in the determination of the eigenvalues of (Formula presented…