Spectrum Of Differential Operator Research Articles

Spectrum of differential operators with elliptic adjoint on a scale of localized Sobolev spaces

In this paper, we provide a complete study of the spectrum of a constant coefficients differential operator on a scale of localized Sobolev spaces, $$H^{s}_{\mathrm{loc}}(I),$$ which are Fréchet spaces. This is quite different from what we find in the literature, where all the relevant results are concerned with spectrum on Banach spaces. Our aim is to understand the behavior of all the three types of spectrum (point, residual, and continuous) and the relation between them and those of the dual operator. The main result we present shows that there is no complex number in the resolvent set of such operators, which suggest a new way to define spectrum if we want to reproduce the classical theorems of the Spectral Theory in Fréchet spaces.

Об изучении спектра семейства дифференциальных операторов, потенциалы которых сходятся к дельта-функции Дирака

Об изучении спектра семейства дифференциальных операторов, потенциалы которых сходятся к дельта-функции Дирака

On the Spectrum of Differential Operators Under Riemannian Coverings

For a Riemannian covering p :M_{2} rightarrow M_{1}, we compare the spectrum of an essentially self-adjoint differential operator D_{1} on a bundle E_{1} rightarrow M_{1} with the spectrum of its lift D_{2} on p^{*}E_{1} rightarrow M_{2}. We prove that if the covering is infinite sheeted and amenable, then the spectrum of D_{1} is contained in the essential spectrum of any self-adjoint extension of D_{2}. We show that if the deck transformations group of the covering is infinite and D_{2} is essentially self-adjoint (or symmetric and bounded from below), then D_{2} (or the Friedrichs extension of D_{2}) does not have eigenvalues of finite multiplicity and in particular, its spectrum is essential. Moreover, we prove that if M_{1} is closed, then p is amenable if and only if it preserves the bottom of the spectrum of some/any Schrödinger operator, extending a result due to Brooks.

SIMILARITY OF OPERATORS ON TENSOR PRODUCTS OF SPACES AND MATRIX DIFFERENTIAL OPERATORS

Let ${\mathcal{H}}=\mathbb{C}^{n}\otimes {\mathcal{E}}$ be the tensor product of a Euclidean space $\mathbb{C}^{n}$ and a separable Hilbert space ${\mathcal{E}}$. Our main object is the operator $G=I_{n}\otimes S+A\otimes I_{{\mathcal{E}}}$, where $S$ is a normal operator in ${\mathcal{E}}$, $A$ is an $n\times n$ matrix, and $I_{n},I_{{\mathcal{E}}}$ are the unit operators in $\mathbb{C}^{n}$ and ${\mathcal{E}}$, respectively. Numerous differential operators with constant matrix coefficients are examples of operator $G$. In the present paper we show that $G$ is similar to an operator $M=I_{n}\otimes S+\hat{D}\times I_{{\mathcal{E}}}$ where $\hat{D}$ is a block matrix, each block of which has a unique eigenvalue. We also obtain a bound for the condition number. That bound enables us to establish norm estimates for functions of $G$, nonregular on the closed convex hull $\operatorname{co}(G)$ of the spectrum of $G$. The functions $G^{-\unicode[STIX]{x1D6FC}}\;(\unicode[STIX]{x1D6FC}>0)$ and $(\ln G)^{-1}$ are examples of such functions. In addition, in the appropriate situations we improve the previously published estimates for the resolvent and functions of $G$ regular on $\operatorname{co}(G)$. Since differential operators with variable coefficients often can be considered as perturbations of operators with constant coefficients, the results mentioned above give us estimates for functions and bounds for the spectra of differential operators with variable coefficients.

Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {Tn(f)}, under suitable assumptions on the associated generating function f. In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {Tn−1(g)Tn(f)}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f in the nonpreconditioned case. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices with a high level of accuracy, with a relatively low computational cost, and with potential application to the computation of the spectrum of differential operators.

On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients

On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients

The conditions for the discreteness of the spectrum of self-adjoint differential operators of even order

In this paper, we consider differential operators of 2nd-order a [ u ]= (-1) k ( a k ( x ) u ( k ) ( x ))( k ), x ∈(0, ∞) whose coefficients a k ( x ) are restricted by powers of e x , and give conditions on the coefficients sufficient to ensure that the spectrum is discrete; next we formulate necessary and sufficient conditions for the discreteness of the spectrum of differential operators whose coefficients a k ( x ) may increase as e αkx as x →∞.

A formula for the spectra of differential operators on graphs

We consider a connected undirected finite graph and a spectral problem generated by the double differentiation of functions on its edges (under usual conditions on the vertices ensuring the self-adjointness of the problem). We introduce, in a standard way, an entire function vanishing at the nonzero eigenvalues of the problem and give an explicit formula for this function, which involves graphs (introduced by V. I. Arnold) generated by a self-mapping of a finite set.

The spectrum of differential operators and square-integrable solutions

We give a comprehensive account of the relationship between the square-integrable solutions for real values of the spectral parameter λ and the spectrum of self-adjoint even order ordinary differential operators with real coefficients and arbitrary deficiency index d and we solve an open problem stated by Weidmann in his well-known 1987 monograph. According to a well-known result, if one endpoint is regular and for some real value of the spectral parameter λ the number of linearly independent square-integrable solutions is less than d, then λ is in the essential spectrum of every self-adjoint realization of the equation. Weidmann extends this result to the two singular endpoint case provided an additional condition is satisfied. Here we prove this result without the additional condition.

The regularized trace of the perturbed Laplace-Beltrami operator on a certain family of manifolds

225 This paper is devoted to calculating the regularized trace of the perturbed Laplace–Beltrami operator on a certain family of manifolds with closed geodesics. After the development of the theory of integral Fourier operators, an overwhelming progress in the study of the spectrum of differential operators on com� pact manifolds with periodic bicharacteristic flow has been achieved. In this theory, the Laplace–Beltrami operator perturbed by the operator of multiplication by a smooth function on manifolds with closed geode� sic flow occupies a special place as the basic model and as the physically most interesting case. Consider the Laplace–Beltrami operator Δ 0 on the sphere S n in R n+1 . By Δ = Δ0 +

Spectral properties of difference and differential operators in weighted spaces

We describe the spectrum of a difference operator (a weighted shift operator). We obtain applications to finding spectra of differential operators in weighted function spaces.

Real-parameter square-integrable solutions and the spectrum of differential operators

We continue to investigate the connection between the spectrum of self-adjoint ordinary differential operators with arbitrary deficiency index d and the number of linearly independent square-integrable solutions for real values of the spectral parameter λ. We show that if, for all λ in an open interval I, there are d linearly independent square-integrable solutions, then there is no continuous spectrum in I. This for any self-adjoint realization with boundary conditions which may be separated, coupled, or mixed. The proof is based on a new characterization of self-adjoint domains and on limit-point (LP) and limit-circle (LC) solutions established in an earlier paper.

On the index and spectrum of differential operators on ℝ^{ℕ}

IfP(x,∂)P(x,\partial )is anr×rr\times rsystem of differential operators onRN\mathbb {R}^{N}having continuous coefficients with vanishing oscillation at infinity, the Cordes–Illner theory ensures thatP(x,∂)P(x,\partial )is Fredholm from(Wm,p)r(W^{m,p})^{r}to(Lp)r(L^{p})^{r}for all or no valuep∈(1,∞).p\in (1,\infty ).We prove that both the index (when defined) and the spectrum ofP(x,∂)P(x,\partial )are independent ofp.p.

The spectrum of differential operators in $H^p$ spaces

The spectrum of differential operators in $H^p$ spaces

Some results on essential spectra of differential operators in Banach spaces

Abstract In this paper we investigate spectral and semi‐Predholm properties of maximum and minimum Puchsian differential operators on Lebesgue spaces on a semi‐axis. These results are applied for determination of various essential spectra and spectrum of ordinary differential operators with polynomial coefficients, which order does not exceed the order of the corresponding derivative.

The transverse geometry of $G$-manifolds and Riemannian foliations

Given a compact Riemannian manifold on which a compact Lie group acts by isometries, it is shown that there exists a Riemannian foliation whose leaf closure space is naturally isometric (as a metric space) to the orbit space of the group action. Furthermore, this isometry (and foliation) may be chosen so that a leaf closure is mapped to an orbit with the same volume, even though the dimension of the orbit may be different from the dimension of the leaf closure. Conversely, given a Riemannian foliation, there is a metric on the basic manifold (an $O(q)$-manifold associated to the foliation) such that the leaf closure space is isometric to the $O(q)$-orbit space of the basic manifold via an isometry that preserves the volume of the leaf closures of maximal dimension. Thus, the orbit space of any Riemannian G-manifold is isometric to the orbit space of a Riemannian $O(q)$-manifold via an isometry that preserves the volumes of orbits of maximal dimension. Consequently, the spectrum of the Laplacian restricted to invariant functions on any $G$-manifold may be identified with the spectrum of the Laplacian restricted to invariant functions on a Riemannian $O(q)$-manifold. Other similar results concerning the spectrum of differential operators on sections of vector bundles over Riemannian foliations and $G$-manifolds are discussed.

Wave Localization in Hydroelastic Systems

The phenomenon of wave localization in hydroelastic systems leads to the strength concentration of radiation fields. The linear method considers the process of localization to be the formation of nonpropagation waves (trapped modes phenomenon). The presence of such waves in the total wave packet points to the existence of mixed natural spectrum of differential operators describing the behaviour of hydroelastic systems. The problem of liquid and oscillating structure interaction caused by the trapped modes phenomenon has been solved (membranes, dies). The interaction of the liquid and elastic structures with inclusions can lead to localized mode formation. In the case of solitary wave motion in nonlinear elastic media, contacting with the liquid, these solitons can be interpreted as “moving inclusions”. The analytical solution for solitary waves has been found. If the soliton speed v0 is more than the velocity of sound c0 in the liquid, the solitary waves strongly slow down. If c0 is close to v0, then a resonance can be observed and solitons move without any resistance. If the soliton speed is less than c0, the solitary wave slow-down is negligible, compared to the case v0 > c0.

Cavalieri conditions for a k-dimensional radon transformation

I. R. Oinarov and A. P. Stikharnyi, "A criterion for the discreteness of the spectrum of the Sturm-Liouville difference operator," in: Ninth Republican Intercollegiate Scientific Conference on Mathematics and Mechanics. Abstracts of Reports. Part I. Mathematics [in Russian], Alma-Ata (1989), p. 33. 2. E. S. Smailov, "Difference imbedding theorems for Sobolev spaces with weights, and their applications," Dokl. Akad. Nauk SSSR, 270, No. i, 52-55 (1983). 3. A. T. Bulabaev and L. M. Mustafina, "Difference imbedding theorems for weighted Sobolev spaces," in: All-Union School for Young Scientists. Functional Methods in Applied Mathematics and Mathematics Physics. Abstracts of Reports. Part II [in Russian], Tashkent (1988), pp. 121-122. 4. R. Oinarov and M. Oteibaev, "A criterion for the discreteness of the spectrum of the general Sturm--Liouville operator, and imbedding theorems connected with it," Differents. Uravn., 24, No. 4, 584-591 (1988). 5. E. T. Sawyer, "A weighted inequality and eigenvalue estimates for Schrodinger operators," Indiana Univ. Math. J., 35, No. i, 1-28 (1986). 6. R. Oinarov, "On the denseness of compactly supported functions in weighted spaces, and on weighted inequalities," Dokl. Akad. Nauk SSSR, 303, No. 3, 559-563 (1988). 7. K. T. Mynbaev and M. O. Otelbaev, Weighted Functional Spaces and the Spectra of Differential Operators [in Russian], Nauka, Moscow (1988). 8. A. Mo Molchanov, "On conditions for discreteness of the spectrum of self-adjoint differential equations of the second order," Tr. Mosk. Mat. Obshch., ~, 169-200 (1953). 9. Functional Analysis. Mathematics Handbook Library [in Russian], Nauka, Moscow (1972).

On the exponential behaviour of eigenfunctions and the essential spectrum of differential operators

SynopsisThe paper deals with the differential equationon [ 0, ∞) Where λ>0 and the coefficients qm are complex-valued with qn continuous and non-zero, w is positive and continuous and qm for m = 0, 1,…, n − 1. In the first part of the paper the exponential behaviour of any solution of (*) is given in terms of a function ρ(λ) which is roughly the distance of λ from the essential spectrum of a closed, densely denned linear operator T generated by T+ in L2(0, ∞ w). Next, estimates are obtained for the solutions in terms of the coefficients in (*). When the latter results are compared with the estimates established previously in terms of ρ(λ), bounds for ρ(λ) are obtained. From the general result there are two kinds of consequences. In the first, criteria for ρ(λ) = 0 for all All λ > 0 are obtained; this means that [0, ∞) lies in the essential spectrum of T in appropriate circumstances. The second type of consequence concerns bounds of the form ρ(λ) = O(λr) for λ → ∞ and r<1.

The continuous spectrum of differential operators

The continuous spectrum of differential operators