Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices

We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter λ and on the vector of real physical parameters p. We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of p. Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD α ²-dynamo and circular string demonstrates the efficiency and applicability of the approach.     

Conservation and bifurcation of an invariant torus of a vector field

We consider small perturbations with respect to a small parameter ε≥0 of a smooth vector field in ℝ^n+m possessing an invariant torusT ^m. The flow on the torusT ^m is assumed to be quasiperiodic withm basic frequencies satisfying certain conditions of Diophantine type; the matrix Ω of the variational equation with respect to the invariant torus is assumed to be constant. We investigate the existence problem for invariant tori of different dimensions for the case in which Ω is a nonsingular matrix that can have purely imaginary eigenvalues. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 34–44, January, 1997. Translated by S. K. Lando     

Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators

We consider quite general h-pseudodifferential operators on R ⁿ with small random perturbations and show that in the limit h → 0 the eigenvalues are distributed according to a Weyl law with a probabality that tends to 1. The first author has previously obtained a similar result in dimension 1. Our class of perturbations is different.     

Irreducibility and Mixed Boundary Conditions

In this paper we consider positive semigroups on L^p(Ω) generated by elliptic operators A subject to mixed Dirichlet-Neumann boundary conditions on non-smooth domains Ω. We show in particular that these semigroups as well as those generated by multiplicative perturbations bA of A are irreducible, provided b ∈ L^∞(Ω) is real and satisfies b ≥ δ for some δ > 0. In memoriam Helmut H. Schaefer     

On the mixing time of a simple random walk on the super critical percolation cluster

 We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Z ^d . We show that for d≥2 and p>p _c (Z ^d ), the mixing time of simple random walk on the largest cluster inside is Θ(n ² ) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovàsz-Kannan average conductance method. This is the first non-trivial application of this method which yields a tight result. Received: 16 December 2001 / Revised version: 13 August 2002 / Published online: 19 December 2002     

Generating Uniform Random Vectors in Z p k : The General Case

We consider the rate of convergence of the Markov chain X _n+1=A X _n +B _n (mod p), where A is an integer matrix with nonzero eigenvalues, and {B _n } _n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Q ^k invariant under A. If for all eigenvalues λ _i of A, then n=O((ln p)²) steps are sufficient and n=O(ln p) steps are necessary to have X _n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λ _i that are roots of positive integer numbers, |λ ₁|=1 and |λ _i |>1 for all , then O(p ²) steps are necessary and sufficient.      

Holomorphic functional calculus of Hodge-Dirac operators in L p

We study the boundedness of the H ^∞ functional calculus for differential operators acting in L ^p (R ⁿ ; C ^N ). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For non-constant coefficients, we extend our recent results for the L ^p theory of the Kato square root problem to the more general framework of Hodge-Dirac operators with variable coefficients Π _B as treated in L ²(R ⁿ ; C ^N ) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that Π _B has a bounded H ^∞ functional calculus, in terms of randomized boundedness conditions of its resolvent. This allows us to deduce stability under small perturbations of this functional calculus.     

Concentrating solutions for the Hénon equation in ℝ2

We consider the boundary value problem Δu+⋎x⋎^2α u ^p =0, α>0, in the unit ballB with homogeneous Dirichlet boundary condition andp a large exponent. We find a condition which ensures the existence of a positive solutionu _p concentrating outside the origin atk symmetric points asp goes to +∞. The same techniques lead also to a more general result on general domains. In particular, we find that concentration at the origin is always possible, provided α⊄IN. The first author is supported by M.U.R.S.T., project “Variational methods and nonlinear differential equations” and a PIMS Postdoctoral Fellowship. The second author is supported by M.U.R.S.T., project “Metodi variazionali e topologici nello studio di fenomeni non lineari.” The third author is supported by an Earmarked Grant from RGC of HK.     

Generation of analytic semigroups in theL p topology by elliptic operators inR n

Strongly elliptic differential operators with (possibly) unbounded lower order coefficients are shown to generate analytic semigroups of linear operators onL ^p(R ⁿ ), 1≦p≦∞. An explicit characterization of the domain is given for 1<p<∞. An application to parabolic problems is also included. This work has been partially supported by the Research Funds of the Ministero della Pubblica Istruzione. The authors are members of GNAFA (Consiglio Nazionale delle Ricerche).     

Perturbed boundary eigenvalue problem for the Schrödinger operator on an interval

A perturbed two-parameter boundary value problem is considered for a second-order differential operator on an interval with Dirichlet conditions. The perturbation is described by the potential μ⁻¹ V((x − x ₀)ɛ⁻¹), where 0 < ɛ ≪ 1 and μ is an arbitrary parameter such that there exists δ > 0 for which ɛ/μ = o(ɛ^δ). It is shown that the eigenvalues of this operator converge, as ɛ → 0, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed.     

Operator Matrices as Generators of Cosine Operator Functions

We introduce an abstract setting that allows to discuss wave equations with time-dependent boundary conditions by means of operator matrices. We show that such problems are well-posed if and only if certain perturbations of the same problems with homogeneous, time-independent boundary conditions are well-posed. As applications we discuss two wave equations in L^p(0, 1) and in L²(Ω) equipped with dynamical and acoustic-like boundary conditions, respectively.     

Evaluation of Chebyshev pseudospectral methods for third order differential equations

When the standard Chebyshev collocation method is used to solve a third order differential equation with one Neumann boundary condition and two Dirichlet boundary conditions, the resulting differentiation matrix has spurious positive eigenvalues and extreme eigenvalue already reaching O(N ⁵ for N = 64. Stable time-steps are therefore very small in this case. A matrix operator with better stability properties is obtained by using the modified Chebyshev collocation method, introduced by Kosloff and Tal Ezer [3]. By a correct choice of mapping and implementation of the Neumann boundary condition, the matrix operator has extreme eigenvalue less than O(N ⁴. The pseudospectral and modified pseudospectral methods are implemented for the solution of one-dimensional third-order partial differential equations and the accuracy of the solutions compared with those by finite difference techniques. The comparison verifies the stability analysis and the modified method allows larger time-steps. Moreover, to obtain the accuracy of the pseudospectral method the finite difference methods are substantially more expensive. Also, for the small N tested, N ⩽ 16, the modified pseudospectral method cannot compete with the standard approach. This revised version was published online in August 2006 with corrections to the Cover Date.     

New boundary element formulas for the biharmonic equation

We propose two new boundary integral equation formulas for the biharmonic equation with the Dirichlet boundary data that arises from plate bending problems in ℝ². Two boundary conditions, u and ∂u/∂n, usually yield a 2 × 2 non-symmetric matrix system of integral equations. Our new formulas yield scalar integral equations that can be handled more efficiently for theoretical and numerical purposes. In this paper we supply complete ellipticity and solvability analyses of our new formulas. Numerical experiments for simple Galerkin methods are also provided. This revised version was published online in June 2006 with corrections to the Cover Date.     

The computation of isotropic vectors

We describe algorithms to compute isotropic vectors for matrices with real or complex entries. These are unit vectors b satisfying b ^* Ab = 0. For real matrices the algorithm uses only the eigenvectors of the symmetric part corresponding to the extreme eigenvalues. For complex matrices, we first use the eigenvalues and eigenvectors of the Hermitian matrix K = (A − A ^*)/2i. This works in many cases. In case of failure we use the Hermitian part H or a combination of eigenvectors of H and K. We give some numerical experiments comparing our algorithms with those proposed by R. Carden and C. Chorianopoulos, P. Psarrakos and F. Uhlig.     

Some questions in the theory of inverse problems for differential operators

For the two operatorsLy=y ⁿ +σ _k=0 ⁿ⁻² p _k (x)y( ^k ) and Ry=yⁿ+σ _k=0 ⁿ⁻² p_k(x)y(^k) with a common set of boundary conditions we establish a connection between p_k(x) and P_k(x) in the case where the weight numbers coincide and a finite number of the eigenvalues do not coincide, in terms of the eigenfunctions of these operators corresponding to the noncoincident eigenvalues and the derivatives of these functions. This enables us to recover the operator L from the operator R by solving a system of nonlinear ordinary differential equations. For Sturm-Liouville operators an analogous relation is proved for the case where infinitely many eigenvalues do not coincide. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 151–160, February, 1977. I wish to express my thanks to my scientific adviser V. A. Sadovnich.     

Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in <Emphasis Type="Italic">l</Emphasis><Superscript>2</Superscript> by the use of finite submatrices

We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l ²(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J _n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].     

Regularity of discs attached to a submanifold ofC n

Letp be an analytic disc attached to a generating CR-submanifoldM of C ⁿ . It is proved that some recently introduced conditions onp andM which imply that the family of all smallC ^α holomorphic perturbations ofp alongM is a Banach submanifold of (A^α(D))ⁿ are equivalent. These conditions are given in terms of the partial indices of the discp attached toM and “holomorphic sections” of the conormal bundle ofM along p(∂D). Also, a sufficient geometric conditionon p andM is given so that the family of all smallC ^α holomorphic perturbationsof p alongM, fixed at some boundary point, is a Banach submanifold of (A ^α (D))ⁿ.     

The Upper Bounds of Arbitrary Eigenvalues for Uniformly Elliptic Operators with Higher Orders

Let Ωbelong to R^m （m≥ 2） be a bounded domain with piecewise smooth and Lipschitz boundary δΩ Let t and r be two nonnegative integers with t ≥ r ＋ 1. In this paper, we consider the variable-coefficient eigenvalue problems with uniformly elliptic differential operators on the left-hand side and （-Δ）^T on the right-hand side. Some upper bounds of the arbitrary eigenvalue are obtained, and several known results are generalized.     

Irrationality of linear combinations of eigenvectors

A givenn ×n matrix of rational numbers acts onC ^π and onQ ^π. We assume that its characteristic polynomial is irreducible and compare a basis of eigenvectors forC ^π with the standard basis forQ ^π. Subject to a hypothesis on the Galois group we prove that vectors from these two bases are as independent of each other as possible.     

Hard Squares for z = �C1

The hard square model in statistical mechanics has been investigated for the case when the activity z is −1. For cyclic boundary conditions, the characteristic polynomial of the transfer matrix has an intriguingly simple structure, all the eigenvalues x being zero, roots of unity, or solutions of x ³ = 4cos²(πm/N). Here we tabulate the results for lattices of up to 12 columns with cyclic or free boundary conditions and the two obvious orientations. We remark that they are all unexpectedly simple and that for the rotated lattice with free or fixed boundary conditions there are obvious likely generalizations to any lattice size.