Operator In L2 Research Articles

Asymptottcs of singular values of volterra integral operators

The singular values sn(K) of the Volterra integral operator in L2(0,1) are proved to have the asymptotics when and h(x,t) is a smooth enough function with respect to the variable x and square integrate. The asymptotics of the singular values of the fractional integral operator (the case [math004]) is given more precisely by

Spectral properties of operators of the theory of harmonic potential

We classify the points of the spectrum of the operatorsB andB * of the theory of harmonic potential on a smooth closed surfaceS ⊂ ℝ3. These operators give the direct value onS of the normal derivative of the simple layer potential and the double layer potential. We show that zero can belong to the point spectrum of both operators inL 2 (S). We prove that the half-interval [−2π, 2π) is densely filled by spectrum points of the operators for a varying surface; this is a generalization of the classical result of Plemelj. We obtain a series of new spectral properties of the operatorsB andB * on ellipsoidal surfaces.

The Operator (sgn x) d 2 dx 2 is Similar to a Selfadjoint Operator in L 2 ( R )

Krein space operator-theoretic methods are used to prove that the operator (sgn x) d is similar to a selfadjoint operator in the Hilbert space L2 (R) . Let L be a symmetric ordinary differential expression. Spectral properties of the operators associated with the weighted eigenvalue problem Lu = )wu have been studied extensively. When w is positive, this problem leads to a selfadjoint problem in the Hilbert space L2(w). In recent years there has been considerable interest in the case when w changes sign; for a survey see [5] and also [2]. In this case the problem may have nonreal and nonsemisimple spectrum. Since the problem is symmetric with respect to an indefinite scalar product, it is natural to consider the problem in the associated Krein space. The corresponding operator can be studied using the spectral theory of definitizable selfadjoint operators in Krein spaces. For definitions and basic results of this theory see [4]. Let (X, [*I*]) be a Krein space, A a definitizable operator in X, and E the spectral function of A. Of particular interest are the so-called critical points of A where its spectral properties are different from the spectral properties of a selfadjoint operator in Hilbert space. Definitizable operators may have at most finitely many critical points. Significantly different behavior of the spectral function occurs at singular critical points in any neighborhood of which the spectral function is unbounded. The critical points which are not singular are called regular. The simplest class of definitizable operators are positive operators with nonempty resolvent set: an operator A is positive if [Ax, x] > 0, x E -'(A). The spectrum of a positive operator is real; only 0 may be a nonsemisimple eigenvalue; only 0 and x may be critical points. Moreover, if 0 and x are not singular critical points and if 0 is not an eigenvalue, then the operator A is a selfadjoint operator in the Hilbert space (X, [(E(R+) E(R_)) Ii); see [4, Theorem 5.7]. Received by the editors June 15, 1993 and, in revised form, July 1, 1993. 1991 Mathematics Subject Classification. Primary 47B50, 47E05; Secondary 47B25, 34L05. The second author was supported in part by Fond za znanstveni rad Hrvatske. This research was done while the second author was visiting the Department of Mathematics, Western Washington University. Q 1995 American Mathematical Society 0002-9939/95 $1.00 + S.25 per page

On theC *-algebra of singular integral operators with carleman shift on a non-closed ARC

In this paperC*-algebra methods are used to obtain a complete description of the algebra generated by singular integral operators inL2([0, 1]) with Carleman shift and continuous coefficients.

Numerical computation of bounded states for schröudinger operators

AbstractA method developed previously for computing eigenfunctions of one‐dimensional Schrödinger operators is extended to Schrödinger operators in L2(R3N). It is known that in many cases these operators have not a compact resolvent; therefore, the convergence in L2(R3N) of the more used methods for computing the eigenfunctions is not guaranteed. The idea of the present method consists of replacing the eigenvalue problem in L2(R3N) by one corresponding to the system confined into a box Ω with impenetrable walls [Dirichlet problem in L2(Ω)]. It is shown that the eigenfunctions of the unbounded system can be approximated by those of the confined system when the box Ω is expanded. On the other hand, it is proved that the Schrödinger operator associated to the confined system has a compact resolvent and its corresponding sesquilinear form is bounded and elliptic in the Sobolev space W2,10(Ω). These properties guarantee the convergence in L2(Ω) of the standard methods to solve the Dirichlet problem: the Ritz method as well as the finite‐element and finite‐difference methods. Therefore, the eigenfuncions of the unbounded system can be approximated in L2(R3N) by means of the numerical solutions of the Dirichlet problem in L2(Ω) with sufficiently large Ω. This property guarantees the accurate computation of the true expectation values. © 1994 John Wiley & Sons, Inc.

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators II

AbstractThis paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type Ψ with χ χ ϵ ℝ and 0 ≤ γ ≤ 1 in the weighted function spaces B (ℝn, w(x)) and F (ℝn, w(x)) treated in [17]. Furthermore, we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b2(x) b(x, D) b1(x), where b1(x) and b2(x) are appropriate functions and b(x, D) ϵ Ψ. Finally, on the basis of the Birman‐Schwinger principle, we deal with the “negative spectrum” (bound states) of related symmetric operators in L2.

BMO on strongly pseudoconvex domains: Hankel operators, duality and \overline∂-estimates

We study the condition that characterizes the symbols of bounded Hankel operators on the Bergman space of a strongly pseudoconvex domain and show that it is equivalent to B M O BMO plus analytic. (Here we mean the Bergman metric B M O BMO of Berger, Coburn and Zhu.) In the course of the proof we obtain new ∂ ¯ \overline \partial -estimates that may be of independent interest. Some applications include a decomposition of B M O BMO similar to the classical L ∞ + L ∞ ~ {L^\infty } + \widetilde {{L^\infty }} , and two characterizations of the dual of V M O VMO (which is also a predual of B M O BMO ). In addition, we obtain some partial results on the boundedness of Hankel operators in L 1 {L^1} norm.

A condition for nonnegativity of an operator related to the Dirichlet operator

Let L0 be a positive definite closed linear operator with domain of definition D(L0) dense in the Hilbert space H; let(ℋ, Γ1, Γ2) be the positive boundary value space of the operator L0 such that the restriction of L 0 * to ker Γ2 is the Friedrichs extension of the operator L0. We establish a test for nonnegativity of an operator T of the form Ty=L 0 * y+Φ*(Γ1−CΦ)y, y ∈ D(T)= ker(Γ2+Φ), where Φ:H → ℋ and C: ℋ → ℋ are respectively a compact operator and a bounded nonnegative operator.

Singular continuous spectrum in a class of random Schrödinger operators

For a class of random Schrödinger operators in L2(Rd▪ where qj are continuous independent identically distributed bounded random variables and f has a power decay and defined sign, in any energy interval the singular continuous spectrum is either empty or with positive Lebesgue measure. As a consequence, the proof of localization for a class of random but deterministic one-dimensional operators is shifted to showing that the singular continuous spectrum has null Lebesgue measure.

Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space

The problem of realization of nontrivial perturbations supported on thin sets of “codimension” ν in Rn for elliptic operators of order m, when ν≥2m, is formulated as one of construction of the self-adjoint extensions of some symmetric linear relation in an indefinite metric space. The self-adjoint extensions and their resolvents are described. It is found that the same extensions can be obtained as a result of extensions of some symmetric operator in L2 (Rn) with it going out to a larger indefinite metric space. But such an operator is chosen already by the “nonlocal” boundary conditions. Applications to quantum models of point interactions are discussed.

Yang-Baxterization of Spin-1 L Operator11The project supported in part by National Natural Science Foundation of China.

The Yang-Baxterization of L operator is performed for spin-1 model, namely, for given braid group representation (BGR) the x-dependent L(x) operator can be determined where x is the spectral parameter. It is pointed out that in order to establish such an x-dependence besides the usual L± operators in accordance with the Faddeev-Reshetikhin-Takhtajan (FRT) approach an additional operator L0 should be introduced. The explicit expression of L0 is given. This Yang-Baxterization prescription preserves the FRT and the YBE is satisfied by L(x) and R(x).

The indentation of two rigid stamps into an elastic sphere with a spheroidal inclusion

On the basis of the expansion formulas of the vector solutions of the Lame equations in spherical coordinates with respect to the solutions of the Lame equations in oblate spheroidal coordinates and on the basis of their inverse formulas, one solves the problem of the compression of an elastic ball with an absolutely rigid inclusion in the form of an oblate spheroid. The problem is reduced to an infinite system of linear algebraic equations of the second kind with a completely continuous operator in l2. Results of the numerical solution of the infinite system are given and the obtained results are analyzed.

Affine symplectic geometry I: Applications to geometric inequalities

We use the integral geometric formulas in the symplectic space of geodesics of a Riemannian manifold to derive various inequalities of isoperimetric type. We give a sharp lower bound for the area of the minimal bubble spanning a spherical curve in ℝ3. We also present an “inverse Croke inequality” relating the area of the boundary of a complex domain in a Riemannian manifold to the injectivity radius and the volume of the domain. We prove a sharp lower bound for the ground state of the harmonic oscillator operator inL 2(M), whereM is a Hadamard manifold.

THE AFFINE N = 4 YANG–MILLS THEORY

An affine field theory may be constructed by gauging an affine algebra. The momentum integrals of the affine N = 4 Yang–Mills theory are ultraviolet finite but diverge because the sum over states is infinite. If the affine symmetry is broken by postulating a nonvanishing vacuum expectation value for that component of the scalar field lying in the L0 direction, then the theory acquires a linear mass spectrum. This broken theory is ultraviolet finite too, but the mass spectrum is unbounded. If it is also postulated that the mass spectrum has an upper bound (say, the Planck mass), then the resulting theory appears to be altogether finite. The influence of the exotic states has been estimated and, according to the proposed scenario, is negligible below energies at which gravitational interactions become important. The final effective theory has the symmetry of a compact Lie algebra augmented by the operator L0.

Axiomatic foundations of quantum physics: Critiques and misunderstandings. Piron's question-proposition system

Some of the most frequent misconceptions about axiomatic quantum physics are discussed with the aim of clarifying their true significance, taking Piron's approach as conceptual framework. In particular, we deal with the following topics: the wrong identification of Piron's questions and Mackey's questions, and some curious alleged empirical consequences; the role of propositions as suitable equivalence classes of questions, their partial order structure, and the paradoxical consequences of the erroneous assignment to questions of some lattice properties involving propositions; the logical and the empirical purport of some "negative" theorems; the standard Hilbert space model of the theory and the consequent "metaphysical disasters" related to some identifications, which are peculiar of this model. A controversy between Foulis-Piron-Randall and Hadjisavvas-ThieffineMugur-Sch/ichter is analyzed on the basis of the proposed Hilbert space model (in which Piron's questions are realized by Hilbertian "effects," i.e., linear bounded operators F such that �9 which clarify the different point of views. As an example, we treat the unsharp localization operators in L2(N).

Spectrum of a self-adjoint operator in L2 (K), where K is a local field; Analog of the Feynman-Kac formula

Spectrum of a self-adjoint operator in L2 (K), where K is a local field; Analog of the Feynman-Kac formula

Analytic semigroups generated by elliptic operators in L¹ and parabolic equations

Analytic semigroups generated by elliptic operators in L¹ and parabolic equations

Partial *-algebras and Volterra convolution of distribution kernels

The set of distribution kernels is considered as a partial *-algebra when the multiplication of two kernels is defined as their Volterra convolution. The partial *-algebra of distribution kernels corresponding to a certain partial *-algebra of closable operators in L2(Ω) is studied.

Analytic semigroups generated by ultraweak operators

SynopsisLet Ω be a regular open subset of RN. We present an improved generation result for nonvariational operators in L1 (Ω). This result is obtained by studying ultraweak operators and by proving generation of analytic semigroups in Lp(Ω)(l<p≦∞) and in . We also characterise interpolation and extrapolation spaces.

Linear operators in l2 with a universality property

AbstractA collection P of bounded linear operators in l2 is constructed in such a manner that given any separable metric space X, and any countable collection F of continuous self-maps of X, there is a homeomorphism h of X onto a subset of l2 such that for each f ∈ F there is P ∈ P with hf = Ph.While similar results were obtained by Baayen and Dc Groot, our construction makes it possible to impose additional conditions on h (depending on F). For example, if all the members of F are uniformly continuous then h too can be made uniformly continuous.