Family Of Self-adjoint Operators Research Articles

Threshold approximations for the exponential of a factorized operator family with correctors taken into account

In a Hilbert space H \mathfrak H , consider a family of selfadjoint operators (a quadratic operator pencil) A ( t ) A(t) , t ∈ R t\in \mathbb {R} , of the form A ( t ) = X ( t ) ∗ X ( t ) A(t) = X(t)^* X(t) , where X ( t ) = X 0 + t X 1 X(t) = X_0 + t X_1 . It is assumed that the point λ 0 = 0 \lambda _0=0 is an isolated eigenvalue of finite multiplicity for the operator A ( 0 ) A(0) . Let F ( t ) F(t) be the spectral projection of the operator A ( t ) A(t) for the interval [ 0 , δ ] [0,\delta ] . Approximations for F ( t ) F(t) and A ( t ) F ( t ) A(t)F(t) for | t | ≤ t 0 |t| \leq t_0 (the so-called threshold approximations) are used to obtain approximations in the operator norm on H \mathfrak H for the operator exponential exp ⁡ ( − i τ A ( t ) ) \exp (-i \tau A(t)) , τ ∈ R \tau \in \mathbb {R} . The numbers δ \delta and t 0 t_0 are controlled explicitly. Next, the behavior for small ε > 0 \varepsilon >0 of the operator exp ⁡ ( − i ε − 2 τ A ( t ) ) \exp (-i \varepsilon ^{-2} \tau A(t)) multiplied by the “smoothing factor” ε s ( t 2 + ε 2 ) − s / 2 \varepsilon ^s (t^2 + \varepsilon ^2)^{-s/2} with a suitable s > 0 s>0 is studied. The obtained approximations are given in terms of the spectral characteristics of the operator A ( t ) A(t) near the lower edge of the spectrum. The results are aimed at application to homogenization of the Schrödinger-type equations with periodic rapidly oscillating coefficients.

Computing Nodal Deficiency with a Refined Dirichlet-to-Neumann Map

Recent work of the authors and their collaborators has uncovered fundamental connections between the Dirichlet-to-Neumann map, the spectral flow of a certain family of self-adjoint operators, and the nodal deficiency of a Laplacian eigenfunction (or an analogous deficiency associated to a non-bipartite equipartition). Using a refined construction of the Dirichlet-to-Neumann map, we strengthen all of these results, in particular getting improved bounds on the nodal deficiency of degenerate eigenfunctions. Our framework is very general, allowing for non-bipartite partitions, non-simple eigenvalues, and non-smooth nodal sets. Consequently, the results can be used in the general study of spectral minimal partitions, not just nodal partitions of generic Laplacian eigenfunctions.

On the spectrum of operator families on discrete groups over minimal dynamical systems

It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in $$\ell ^p ({\mathbb {Z}})$$ . Here, we generalize this to a large class of bounded linear operator families on Banach-space valued $$\ell ^p$$ -spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are techniques from limit operator theory.

Continuity of the Spectrum of a Field of Self-Adjoint Operators

Given a family of self-adjoint operators $${(A_t)_{t \in T}}$$ indexed by a parameter t in some topological space T, necessary and sufficient conditions are given for the spectrum $${\sigma(A_t)}$$ to be Vietoris continuous with respect to t. Equivalently the boundaries and the gap edges are continuous in t. If (T, d) is a complete metric space with metric d, these conditions are extended to guarantee Hölder continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps.

Approximations of Strongly Continuous Families of Unbounded Self-Adjoint Operators

The problem of approximating the discrete spectra of families of self-adjoint operators that are merely strongly continuous is addressed. It is well-known that the spectrum need not vary continuously (as a set) under strong perturbations. However, it is shown that under an additional compactness assumption the spectrum does vary continuously, and a family of symmetric finite-dimensional approximations is constructed. An important feature of these approximations is that they are valid for the entire family uniformly. An application of this result to the study of plasma instabilities is illustrated.

A uniqueness theorem for higher order anharmonic oscillators

We study for \alpha\in\mathbb R , k \in {\mathbb N} \setminus \{0\} the family of self-adjoint operators -\frac{d^2}{dt^2}+\Bigl(\frac{t^{k+1}}{k+1}-\alpha\Bigr)^2 in L^2(\mathbb R) and show that if k is even then \alpha=0 gives the unique minimum of the lowest eigenvalue of this family of operators. Combined with earlier results this gives that for any k \geq 1 , the lowest eigenvalue has a unique minimum as a function of \alpha .

Local research of manifolds generated by families of self-adjoint operators

We consider V.I. Arnold’s manifold of self-adjoint operators with fixed multiplicity of eigenvalues and K. Uhlenbeck’s manifold of eigenvectors. Our aim is to consider the local analysis and the connection between these manifolds. We present the topological description of the spectrum perturbation problem, specifically the finite-multiple eigenvalue splitting problem. For investigation of manifolds, we use the local diffeomorphism introduced by D. Fujiwara, M. Tanikawa, and Sh. Yukita.

The Kostant–García quantization of the bundle of connections

We shall call quantum states of a principal bundle π : P → M with structure group a semi-simple Lie group G, the elements of certain space \({\mathcal{H}}\) of sections of the adjoint bundle \({\pi _{\mathcal{G}} : ad(J^{1}P) \rightarrow \mathcal{K}}\), associated to the G-bundle of connections \({p : J^{1}P \rightarrow \mathcal{K}}\) . An inner product \({\left\langle , \right\rangle}\) of sections of \({\pi _{\mathcal{G}}}\) is defined for which \({(\mathcal{H}, \left\langle , \right\rangle)}\) is a Hilbert space such that the Gauge group gau(P) of the given bundle represents in a family of self-adjoint operators. This work crystallizes some heuristic considerations, on the unitary representations of Gauge algebras, of Garcia in the already a classical article (J. Differ. Geom. 12, 209–227, 1977).

Singular Spectrum Near a Singular Point of Friedrichs Model Operators of Absolute Type

In \(L_{2} {\left( \mathbb{R} \right)}\) we consider a family of selfadjoint operators of the Friedrichs model: \(A_{m} = {\left| t \right|}^{m} \cdot + V\). Here \({\left| t \right|}^{m} \cdot \) is the operator of multiplication by the corresponding function of the independent variable \(t \in \mathbb{R}\), and \(V\) (perturbation) is a trace-class integral operator with a continuous Hermitian kernel \(v{\left( {t,x} \right)}\) satisfying some smoothness condition. These absolute type operators have one singular point \(t = 0\) of order \(m > 0\). Conditions on the kernel \(v{\left( {t,x} \right)}\) are found guaranteeing the absence of the point spectrum and the singular continuous one of such operators near the origin. These conditions are actually necessary and sufficient. They depend on the finiteness of the rank of a perturbation operator and on the order of singularity \(m\). The sharpness of these conditions is confirmed by counterexamples.

Self-adjoint extensions of the Laplacian and Aharonov–Bohm operators with a potential supported on a circle

Using the definition of a self-adjoint operator, we give the self-adjoint extensions of the Laplacian and Aharonov–Bohm operators with a potential supported on a circle of radius R. This study gives rise to interesting results, namely all self-adjoint extensions are given by a four-parameter family of self-adjoint operators, reproducing a matrix representation for U(2) symmetry.

Multivariate versions of Cochran's theorems

Let E, V be n-, p-dimensional inner product spaces over the real field, let L ( V, E) be the set of all linear maps of V into E, and let Y be a normal random vector in L ( V, E) with mean μ = 0 and covariance Σ Y such that S 1 ⊏ S 2(≠ }0{) is the image set, Im Σ Y of Σ Y , where S 1, S 2 are linear subspaces of E, V, respectively, and ⊏ is the outer product. Let } W i { be a family of self-adjoint operators in L ( E, E). Then (∗): } Y′ W i Y{ is an independent family of Wishart random operators Y′ W i Y with parameter ( m i , Σ, λ i ), each m i > 0 and λ i = 0,” if and only if Im Σ = S 2 and for any distinct i, j∈ I, Σ Y ( W i ⊗ Σ +) Σ Y ( W i ⊗ Σ +) Σ Y = Σ Y ( W i ⊗ Σ +) Σ Y , tr( Σ Y ( W i ⊗ Σ +)) ≠ 0, and Σ Y ( W i ⊗ Σ +) Σ Y ( W j ⊗ Σ +) Σ Y = 0. A necessary and sufficient condition for (∗) is also obtained for the general case where no condition whatever is imposed on ( μ, Σ Y ). This generalizes a recent result of Pavur who considered the case where Σ is nonsingular and each W i is nonnegative definite.

Concavity and convexity of eigenvalues

This paper completes and refines some results of a previous one, concerning differentiability of eigenvectors of a family of self-adjoint operators

Convexity properties of distinguished eigenvalues of certain classes of operators

We prove two convexity results: (1) Let A ( ε ) = A 0 + ε A 1 A(\varepsilon ) = {A_0} + \varepsilon {A_1} be a family of selfadjoint operators in a Krein space with separated spectrum so that the maximum λ _ ( ε ) \lambda \_(\varepsilon ) of the spectrum of negative type of A ( ε ) A(\varepsilon ) is an isolated simple eigenvalue. Then λ _ ( ε ) \lambda \_(\varepsilon ) is convex. (2) Let λ _ ( ε ) \lambda \_(\varepsilon ) be the left distinguished eigenvalue of the generalized eigenvalue problem ( A 0 + ε A 1 ) x = λ B x ({A_0} + \varepsilon {A_1})x = \lambda Bx where A 1 {A_1} and B B are real diagonal matrices and A 0 {A_0} is an irreducible essentially nonnegative matrix. Then λ _ ( ε ) \lambda \_(\varepsilon ) is convex. In both cases λ _ ( ε ) \lambda \_(\varepsilon ) is strictly convex unless it is linear.

A theorem concerning differentiability of eigenvectors and eigenvales with some applications

Let H(ν) be a family of self-adjoint operators, bounded or unbounded, in a separable Hilbert space H. It is assumed that H (ν) as a function of the real parameter ν is differentiable, with respect to the operator norm, in some open interval l 0 and the derivative H′(ν) is uniformly bounded on compact subsets of l 0. Under the additional assumption that an isolated eigenvalue λ(ν) exists and is simple for every v in L it is proved that the corresponding eigenvector x(ν) is strongly differentiable in l 0. The differential equation λ′ (ν) = (H′ (ν)x(ν), x(v)), ‖ x(ν) ‖ =1 that follows immediately generalizes a diferential equation found for the zeros of Bessel functions in ref [2] and can be applied to many other eigenvalue problems. Techniques, known from the theory of Bessel functions, are used to give bounds and qualitative properties of the eigenvalues of several eigenvalue problems for which both the existence and the simplicity of eigenvalues is well-known

Duality in spectral optimization and numerical ranges of a family of self-adjoint operators

One gives results which complement the author's investigation, published in Dokl. Akad. Nauk SSSR,255, No. 4, 777–780 (1980). One establishes the relationship of the fundamental condition ensuring the duality relation between the direct and the dual problems in spectral optimization problems with the geometry of the numerical ranges of certain families of self-adjoint operators.

Spectral representations and density operators for infinite-dimensional homogeneous random fields

A theory of spectral representations and spectral density operators of infinite-dimensional homogeneous random fields is established. Some results concerning the form of the spectral representation are given in the general infinite-dimensional case, while the results pertaining to the density operator are confined to Hilbert space valued fields. The concept of a purely non-deterministic (p.n.d.) field is defined, and necessary and sufficient conditions for the property of p.n.d. are obtained in terms of the spectral density operator. The theory is developed using some isomorphisms induced by families of self-adjoint operators in the linear second order space associated with the field. The method seems to lead to more direct results also in the random process case, and it sheds new light on concepts such as multiplicity of the field and rank of the spectral density operator.

Simultaneous unitary equivalence

Let At be a family of selfadjoint operators obtained from the orthogonal group O(t) by means of At . The spectrum of all elements in the linear span of the At is shown to be a complete set of simultaneous unitary invariants for the family At .

Spectral representations for families of self-adjoint operators

AbstractSpectral theorems and a functional calculus for families of permutable self-adjoint operators in a separable Hilbert space are developed in a form convenient for physical applications. A recent paper of Jauch and Misra is discussed.

Perturbation of Eigenvalues—with an Engineering Application

Introduction. The mathematician has presently at his disposal a wealth of information concerning perturbations of the point spectrum of a closed linear operator acting in a Banach space. The earliest results are due to F. Rellich [1, 2, 3, 4, 5] who between 1936 and 1941 published a series of papers which included consideration of the following situation: Suppose that L( e) is a family of self-adjoint operators having fixed domain in a Hilbert space, H, and depending analytically (the precise meaning of this restriction need not concern us) upon the real parameter, e, for I e I < p. Suppose further that X0 is an isolated eigenvalue of L(0) having u0 as a corresponding eigenvector: