Operator In L2 Research Articles

Deficiency Indices of a Symmetric Ordinary Differential Operator with Infinitely Many Degeneration Points

Let H be the minimal symmetric operator in L2(ℝ) generated by the differential expression (-1)n(c(x)f(n))(n), n ≥ 1, with a real coefficient c(x) that has countably many zeros without finite accumulation points and is infinitely smooth at all points x ∈ ℝ with c(x) ≠ 0. We study the value Def H of the deficiency indices of H. It is shown that DefH=+∞ if infinitely many zeros of c(x) have multiplicities p satisfying the inequality n − 1/2 < p < 2n − 1/2. Our second result pertains to the case in which the set of zeros of c(x) is bounded neither above nor below. Under this condition, Def H = 0 provided that the multiplicity of each zero is greater than or equal to 2n − 1/2. The multiplicities of zeros of c(x) are understood in the paper in a broader sense than in the standard definition.

On Fourier Series in Eigenfunctions of Elliptic Boundary Value Problems

Abstract We consider a general elliptic formally self-adjoint problem in a bounded domain with homogeneous boundary conditions under the assumption that the boundary and coefficients are infinitely smooth. The operator in 𝐿2(Ω) corresponding to this problem has an orthonormal basis {𝑢𝑙} of eigenfunctions, which are infinitely smooth in . However, the system {𝑢𝑙} is not a basis in Sobolev spaces 𝐻𝑡 (Ω) of high order. We note and discuss the following possibility: for an arbitrarily large 𝑡, for each function 𝑢 ∈ 𝐻𝑡 (Ω) one can explicitly construct a function 𝑢0 ∈ 𝐻𝑡 (Ω) such that the Fourier series of the difference 𝑢 – 𝑢0 in the functions 𝑢𝑙 converges to this difference in 𝐻𝑡 (Ω). Moreover, the function 𝑢(𝑥) is viewed as a solution of the corresponding nonhomogeneous elliptic problem and is not assumed to be known a priori; only the right-hand sides of the elliptic equation and the boundary conditions for 𝑢 are assumed to be given. These data are also sufficient for the computation of the Fourier coefficients of 𝑢 – 𝑢0. The function 𝑢0 is obtained by applying some linear operator to these right-hand sides.

ON FAST AND SLOW TIMES IN MODELS WITH DIFFUSION

The linear Kelvin–Voigt operator ℒε is a typical example of wave operator ℒ0 perturbed by higher-order viscous terms as εuxxt. If [Formula: see text] is a prefixed boundary value problem for ℒε, when ε = 0, ℒε turns into ℒ0 and [Formula: see text] into a problem [Formula: see text] with the same initial–boundary conditions of [Formula: see text]. Boundary layers are missing and the related control terms depending on the fast time are negligible. In a same time interval, the wave behavior is a realistic approximation of uε when ε → 0. On the contrary, when t is large, diffusion effects should prevail and the behavior of uε for ε → 0 and t → ∞ should be analyzed. For this, a suitable functional correspondence between the Green functions [Formula: see text] and [Formula: see text] of [Formula: see text] and [Formula: see text] is derived and its asymptotic behavior is rigorously examined. As a consequence, the interaction between diffusion effects and pure waves is evaluated by means of the slow time εt; the main results show that in time intervals as (ε, 1/ε) pure waves are quasi-undamped, while damped oscillations predominate as from the instant t &gt; 1/ε.

On the Density of Discrete Spectra of Sturm-Liouville Singular Operators

and by L0 the minimal operator (see [1]) generated by the expression (1) in L 2 n[0,∞) . Under certain conditions imposed on the matrix A(x) , the operator L0 has deficiency indices equal to (2n, 2n) and it follows that the spectrum of any self-adjoint extension of the operator is discrete. Let L be a real self-adjoint extension, and let λ0 , λ±1 , . . . , λ±k , . . . be the eigenvalues of the operator L arranged in the increasing order of their absolute values. For λ ≥ 0, set N+(L, λ) = ∑ 0≤λk≤λ 1,

On the negative discrete spectrum of a preiodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential

Let Ω ⊂R d be an unbounded domain, periodic along a chosen direction (a waveguide-type domain),P be a self-adjoint elliptic second order operator inL 2(Ω) periodic along the same direction, andV be a real-valued decaying potential. We suppose that the bottom of the spectrum ofP is λ=0 and study the asymptotic behaviour of the number of negative eigenvalues of the opeatorP−aV as the parameter α tends to +∞. We show that typically the Weyl asymptotic law for this quantity is violated and find a substitute for this law.

Perturbation of orthonormal bases inL 2-spaces

The paper improves and generalizes a classical result from Paley and Wiener in their book on Fourier transforms. Paley and Wiener gave conditions on functionsh n that imply that the sequence (1+h n (x))e inx is a Riesz basis forL 2[−π,π]. These conditions involve theL 2-norm of the second derivativesh ″ . The new results replace the differential operatory→y″ by more general differential operators inL 2-spaces, in particular, by the Hermite differential operator inL 2(R), ande inx by arbitrary orthonormal bases.

Nonexpansive Periodic Operators in l1 with Application to Superhigh-Frequency Oscillations in a Discontinuous Dynamical System with Time Delay

We prove that the iterates of certain periodic nonexpansive operators in l1 uniformly converge to zero in l∞ norm. As a by-product we show that, for any solution x(t) of the equation x(t)= −sign(x(t-1))f(x()), t≥0, x|[−1,0]∈C[−1,0] where f:ℝ→(−1, 1) is locally Lipschitz, the number of zeros of x(t) on any unit interval becomes finite after a period of time, with the single exception of the case f(0)=0 and x(t)≡0.

A Formula for the First Regularized Trace of a Perturbed Laplace–Beltrami Operator

The theory of regularized traces is well developed for problems induced by ordinary differential expressions. It was shown in the fundamental paper [1] that the derivation of trace formulas can be reduced to studying zeros of entire functions possessing a specific asymptotic structure related to the particular form of the fundamental solution system of the differential equation. The case of problems induced by partial differential operators is much harder. This is primarily due to the complicated structure of the spectrum. The asymptotic spectrum distribution was obtained in [2] for the operator −∆+V with smooth odd potential V on the sphere S. However, the results of [2] do not allow finding regularized trace formulas. Further substantial progress in this problem was made in [3], where a formula for the first regularized trace was obtained for the first time for the operator −∆ + V with odd smooth potential V . Later, the method of [4] was used in [5] for finding the first regularized trace for an odd potential with n = 2 and in [6] for deriving a regularized trace formula for the operator −∆+V on compact symmetric spaces of rank 1. However, the condition that the potential V is C∞ was substantially used in all these papers. In the present paper, we derive a formula for the first trace of the operator −∆ +V on S in the case of a finite smoothness of the potential V and without the assumption that V is odd. If the potential is odd, then formula (1) coincides with the corresponding formula in [5]. Let L0 = −∆, and let V be the operator of multiplication by a real smooth function in L2 (S), where S is the two-dimensional sphere. By L we denote the operator L0 +V . Further, let R0(z) = (L0 − zE) and R(z) = (L− zE)−1; let λn = n(n+ 1) and μ n , n = 0, 1, . . . , −n ≤ i ≤ n, be the eigenvalues of the operators L0 and L, respectively, and let f (i) n = Y (i) n (φ, θ) be the orthonormal spherical functions, that is, the eigenfunctions of the operator L0. The following assertion is the main result of the present paper.

Conditions for Convergence of Biorthogonal Expansions of Functions on a Closed Interval

d = 0, . . . , 2n−1, where either μ0 = 1 (Problem 1) or μ0 = cλ2n−1 if |λ| ≥ 1 and μ0 = c = const 6= 0 if |λ| 0 and choose an arbitrary numerical sequence {λk}k=1, λk ∈ S1 = {λ ∈ C : Reλ ≥ 0, | Imλ| ≤ γ0}, and an arbitrary system {uk(x)} of root functions of the operator L corresponding to the spectral parameters {λk} and satisfying the following conditions A: (1) {uk} is a closed minimal system in L 0(G), and the biorthogonal system {vk} satisfies ‖uk‖r0 ‖vk‖r′ 0 ≤ c1 for all k; (2) ∑ 0≤|λk|−λ≤1 1 ≤ c2 for all λ ≥ 0; (3) ∥∥∥m−1 uk ∥∥∥ r0 ≤ c3αλ ∥∥∥muk∥∥∥ r0 , m = 1, . . . ,mk, where αλ = |λk| for Problem 1 and αλ = 1 for Problem 2; (4) ‖uk‖∞ ≤ c4 ‖uk‖r0 for all k, where ci, i = 1, 2, 3, 4, are some constants independent of λk. Let f(x) ∈ L 0(G), and let σλ(x, f) = ∑ |λk|≤λ fkuk(x), λ > 0, fk ≡ (f, vk), be a partial sum of the biorthogonal expansion of f(x). By Sλ(x, f) we denote the corresponding partial sum of the trigonometric Fourier series of f(x) treated as an orthogonal expansion of f for the operator L0 determined by L0u = u′′, u ∈ D2, u(0) = u(1), and u′(0) = u′(1). We assume that

Spectral functions of self-adjoint and symmetric multiplication operators inL 2(X, μ)-spaces

Explicit formulas are derived for the spectral function of double multiplication operator containing a multiplicative evolution inL2(X, μ)-space and a convolution-type operator inL2(ℝn)-spaces. Symmetric convolution and multiplication operators are considered inL2(X, μ) andL2(ℝn)-spaces.

On the essential spectrum of a differentially rotating star in the axisymmetric case

A mixed-order system of singular partial differential equations occurring in the study of rotating stars is considered. It is shown that the associated operator L0 is closable and that the essential spectrum of its closure L coincides with the essential spectrum of a bounded operator. Finally, some parts of the essential spectrum of L are determined explicitly.

THE FIELD OF A VERTICAL MAGNETIC DIPOLE IN THE PRESENCE OF A SPHERE WITH A CIRCULAR APERTURE

We consider diffraction of an arbitrary monochromatic electromagnetic wave by a perfectly conducting, infinitesimally thin sphere with a circular aperture. The initial boundary-value problem is reduced to a system of linear algebraic equations of the second kind in the form (I+H)x=b, where x,b∈l2 and H is a compact operator in l2 (I is the unit operator in l2). We calculate the spectrum of natural frequencies of a sphere with a circular aperture that correspond to axisymmetric modes of magnetic type. The influence of the resonance regimes (in particular, the mode coupling) of the considered structure on its scattering properties is analyzed.

On Form-Sum Approximations of Singularly Perturbed Positive Self-adjoint Operators

We discuss singular perturbations of a self-adjoint positive operator A in Hilbert space H formally given by AT=A+T, where T is a singular positive operator (singularity means that KerT is dense in H). We prove the following result: if T is strongly singular with respect to A in the sense that KerT is dense in the Hilbert space H1(A)=D(A1/2) equipped by the graph-norm, then any suitable approximation by positive operators, Tn→T, gives a trivial result, i.e., ATn→A in the strong resolvent sense, where ATn is defined as a form-sum of A and Tn. A corresponding statement is true for operators T, Tn of finite rank which are not necessarily positive. This can be considered as an abstract version of the well known result for the perturbation by a point interaction of the Laplace operator in L2(R3). In the more general case, where the singular operator T has a nontrivial regular component Tr in H1(A), we prove that ATn→ATr in the strong resolvent sense. We give applications to the case of perturbations of the Laplace operator by a positive Radon measure with a nontrivial singular component.

Schrödinger Operators in L2([formula omitted]) with Pointwise Localized Potential

Schrödinger Operators in L2([formula omitted]) with Pointwise Localized Potential

Algebras of Pseudodifferential Operators in L2 Given by Smooth Measures on Hilbert Spaces

AbstractPseudodifferential operators with symbols on a Hilbert phase space are defined as symmetric operators in L2 given by a smooth measure. The main formulae of symbolic calculus are proved in this context.

Interwining Operators for Vertex Representations of Toroidal Lie Algebras

The construction of the vertex representation of the toroidal Lie algebra τ[n] depends on the way of labelling the points in the dual n of the torus Tn. Thus there is a built-in symmetry of the vertex representation with respect to the symmetry of Zn. In conjunction with this, the energy operator L0 gives rise to intertwining operators which reflect the symmetry of the vertex representation with respect to S1 action on τ.

Uniqueness of Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ d

Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ d are studied. Introducing a linear operator L0 defined on a space of smooth local functions, we show the uniqueness of Dirichlet forms associated with self adjoint Markovian extensions of L0. We also discuss the ergodicity of the reversible process associated with the Dirichlet form.

Un critère de rationalité provenant de la géométrie non commutative

We prove a conjecture of A. Connes, which gives a rationality criterion for elements of the closure of ℂΓ (Γ a free group) in the space of bounded operators in l 2(Γ). We show that this criterion applies also to the ring of Malcev-Neumann series on Γ.

Asymptotic of the density of states for the Schrödinger operator with periodic electromagnetic potential

For the Schrödinger operator in L2(Rn), n&amp;gt;1, with C∞ periodic electromagnetic potential, we give an asymptotic formula of the integrate density of states of the form N(μ)=anμn/2+O(μ(n−2+ε)/2), ∀ε&amp;gt;0. When n=2, this estimate enables us to prove the finiteness of gaps in the spectrum.

Multipliers for semigroups

Let L be a positive invertible self-adjoint operator in L2(X;C). Using transference methods for locally bounded groups of operators we obtain multipliers for the group of complex powers Liu on vector-valued Lebesgue spaces. Using a Mellin inversion formula, we derive a sufficient condition for a function to be a multiplier of the semigroup e-tL on Lp(X;E) where E is a UMD Banach space and 1&lt;p&lt;∞.