On the divergence free eigenfunctions of the magnetostatic integral operator

For the eigenvalues from (−1,1) it is shown that the divergence free eigenfunctions of the adjoint magnetostatic integral operator can be explicitly obtained from the eigenfunctions of the adjoint electrostatic integral operator to the same eigenvalue.     

Prescribing eigenvalues of the Dirac operator

In this note we show that every compact spin manifold of dimension ≥3 can be given a Riemannian metric for which a finite part of the spectrum of the Dirac operator consists of arbitrarily prescribed eigenvalues with multiplicity 1.     

On the eigenvalues of the fredholm operator

We prove that if ω(t, x, K ₂ ^(m) )?c(x)ω(t) for allxε[a, b] andx ε [0,b-a] wherec ∈L ¹(a, b) and ω is a modulus of continuity, then λ _n =O(n ^?m-1/2ω(1/n)) and this estimate is unimprovable.     

Estimates for eigenvalues of the Paneitz operator

For an n  -dimensional compact submanifold Mⁿ$M^n$ in the Euclidean space R^N${\mathbf{R}}^N$, we study estimates for eigenvalues of the Paneitz operator on Mⁿ$M^n$. Our estimates for eigenvalues are sharp.     

On the eigenvalues of orderbounded integral operators

Operators in Banach function spaces which together with their adjoints are orderbounded, have 4th power summable eigenvalues. In general this is best possible. The optimal summability exponent of the eigenvalues of such maps in general is between 2 and 4 and depends on the p-convexity and q-concavity index of the function lattice.     

Purely imaginary eigenvalues of operator semigroups

For aC ₀-contraction semigroup (S(t)) _t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e ^iλt x, t≥0, for some λ∈ℝ, provided lim _t→∞ |<S(t)x,x ^* >|=|<x,x ^* >| for allx ^*∈X^*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.     

Accurate computation of higher Sturm-Liouville eigenvalues

Summary The computation of eigenvalues of regular Sturm-Liouville problems is considered. It is shown that a simple step-dependent linear multistep method can be used to reduce the error of the orderk ⁴ h ² of the centered finite difference estimate of thek-th eigenvalue with uniform step lengthh, to an error of orderkh ². By an appropriate minimization of the local error term of the method one can obtain even more accurate results. A comparison of the simple correction techniques of Paine, de Hoog and Anderssen and of Andrew and Paine is given. Numerical examples demonstrate the usefulness of this correction even for low values ofk.Research Director of the National Fund for Scientific Research (N.F.W.O. Belgium)     

The distribution of eigenvalues of the Dirac operator

The Dirac system is studied on (-, ). Asymptotic formulas are obtained of the distribution for both positive, and negative eigenvalues. The asymptotic formulas, established in Theorem 1, are essentially different from formulas obtained by Sargsyan [1], and permit asymptotic formulas to be written for the distribution of positive (negative) eigenvalues, even in those cases when the negative (positive) spectrum is continuous, if appropriate conditions hold on the potential.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 843–852, December, 1973.In conclusion the author thanks R. S. Ismagilov and B. M. Levitan for valuable instructions and attention to this work.     

Lower bounds for eigenvalues of the Dirac-Witten operator

We get optimal lower bounds for the eigenvalues of the Dirac-Witten operator of compact(with or without boundary) spacelike hypersurfaces of Lorentian manifold satisfying certain conditions,just in terms of the mean curvature and the scalar curvature and the spinor energy-momentum tensor. In the limiting case,the spacelike hypersurface is either maximal and Einstein manifold with positive scalar curvature or Ricci-flat manifold with nonzero constant mean curvature.     

Universal bounds for eigenvalues of the biharmonic operator

In this paper, we study the eigenvalues of the clamped plate problem:

     

A note concerning the integral equations for magnetostatic fields

Zeitschrift für angewandte Mathematik und Physik -     

Extrinsic estimates for eigenvalues of the Dirac operator

For a compact spin manifold M isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the square of the Dirac operator, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues. Since the unit sphere and the projective spaces admit the standard embedding into Euclidean spaces, we also obtain the corresponding results for their compact spin submanifolds.     

Analytic continuation of eigenvalues of the Lamé operator

Eigenvalues of the Lamé operator are studied as complex-analytic functions in period τ of an elliptic function. We investigate the branching of eigenvalues numerically and clarify the relationship between the branching of eigenvalues and the convergent radius of a perturbation series.