Self-adjoint Differential Operators Research Articles

Notes on the non self-adjoint elliptic differential operators

Notes on the non self-adjoint elliptic differential operators

Relationships among Three Multiplicities of a Differential Operator’s Eigenvalue

In this paper, the algebraic, geometric and analytic multiplicities of an eigenvalue for linear differential operators are defined and classified. The relationships among three multiplicities of an eigenvalue of the linear differential operator are given, and a fundamental fact that the algebraic, geometric and analytic multiplicities for any eigenvalue of self-adjoint differential operators are equal is proven.

Spectral theoretic characterization of the massless Dirac operator

We consider an elliptic self-adjoint first-order differential operator acting on pairs (2-columns) of complex-valued half-densities over a connected compact three-dimensional manifold without boundary. The principal symbol of our operator is assumed to be trace-free. We study the spectral function which is the sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive λ. We derive an explicit two-term asymptotic formula for the spectral function as λ to + ∞, expressing the second asymptotic coefficient via the trace of the subprincipal symbol and the geometric objects encoded within the principal symbol-metric, torsion of the teleparallel connection and topological charge. We then address the question: is our operator a massless Dirac operator on half-densities? We prove that it is a massless Dirac operator on half-densities if and only if the following two conditions are satisfied at every point of the manifold: (a) the subprincipal symbol is proportional to the identity matrix; and (b) the second asymptotic coefficient of the spectral function is zero.

Convergence rates and explicit error bounds of Hill’s method for spectra of self-adjoint differential operators

We present the convergence rates and the explicit error bounds of Hill's method, which is a numerical method for computing the spectra of ordinary differential operators with periodic coefficients. This method approximates the operator by a finite dimensional matrix. On the assumption that the operator is selfadjoint, it is shown that, under some conditions, we can obtain the convergence rates of eigenvalues with respect to the dimension and the explicit error bounds. Numerical examples demonstrate that we can verify these conditions using Gershgorin's theorem for some real problems. Main theorems are proved using the Dunford integrals which project an eigenvector to the corresponding eigenspace.

Square function and maximal function estimates for operators beyond divergence form equations

We prove square function estimates in L 2 for general operators of the form B 1 D 1 + D 2 B 2, where D i are partially elliptic constant coefficient homogeneous first-order self-adjoint differential operators with orthogonal ranges, and B i are bounded accretive multiplication operators, extending earlier estimates from the Kato square root problem to a wider class of operators. The main novelty is that B 1 and B 2 are not assumed to be related in any way. We show how these operators appear naturally from exterior differential systems with boundary data in L 2. We also prove non-tangential maximal function estimates, where our proof needs only off-diagonal decay of resolvents in L 2, unlike earlier proofs which relied on interpolation and L p estimates.

Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation

The non-commutative harmonic oscillator (NcHO) is a special type of self-adjoint ordinary differential operator with non-commutative coefficients. In the present note, we aim to provide a reasonable criterion that derives the simplicity of the lowest eigenvalue of NcHO. It actually proves the simplicity of the lowest eigenvalue for a large class of structure parameters. Moreover, this note describes a certain equivalence between the spectral problem of the NcHO (for the even parity) and existence of holomorphic solutions of Heun’s ordinary differential equations in a complex domain. The corresponding Riemann scheme allows us to give another proof to the criterion.

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 →∞.

Factorization of self-adjoint ordinary differential equations

This paper deals with the factorization of self-adjoint differential operators L(2n)=1ρdndxnρβndndxn, and their spectral type differential equations. Sufficient conditions of factorization are reported. A large class of differential operators and equations that can be factorized is obtained. The factorizations of fourth- and sixth-order operators and equations are explicitly given. A particular fourth-order spectral type differential equation in which ρ(x)=(1-x)p(1+x)q, p⩾1,q⩾1, is considered. Its general solution is obtained in terms of hypergeometric functions. As application, the natural frequencies and mode shapes of mechanical transverse vibrations of a nonuniform structure are found.

On Ambarzumyan-type theorems

Generalizations of the classical Ambarzumyan theorem are provided for wide classes of self-adjoint differential operators with arbitrary self-adjoint boundary conditions: scalar Sturm–Liouville operators, higher-order differential operators, matrix Sturm–Liouville operators and operators on spatial networks.

A sinc-Gaussian technique for computing eigenvalues of second-order linear pencils

The sinc-Gaussian sampling technique derived by Qian (2002) establishes a sampling technique which converges faster than the classical sampling technique. Schmeisser and Stenger (2007) studied the associated error analysis. In the present paper we apply a sinc-Gaussian technique to compute the eigenvalues of a second-order operator pencil of the form Q−λP approximately. Here Q and P are self-adjoint differential operators of the second and first order respectively. In addition, the eigenparameter appears in the boundary conditions linearly. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than the classical sinc-method. This is confirmed via worked examples which are given at the end of the paper with comparisons with the classical sinc-method.

Approximate diagonalization of variable-coefficient differential operators through similarity transformations

Approaches to approximate diagonalization of variable-coefficient differential operators using similarity transformations are presented. These diagonalization techniques are inspired by the interpretation of the Uncertainty Principle by Fefferman, known as the SAK Principle, that suggests the location of eigenfunctions of self-adjoint differential operators in phase space. The similarity transformations are constructed using canonical transformations of symbols and anti-differential operators for making lower-order corrections. Numerical results indicate that the symbols of transformed operators can be made to closely resemble those of constant-coefficient operators, and that approximate eigenfunctions can readily be obtained.

ON THE SINGULARITIES OF THE ZETA AND ETA FUNCTIONS OF AN ELLIPTIC OPERATOR

Let P be a self-adjoint elliptic operator of order m > 0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form [Formula: see text], where k ranges over all nonzero integers ≤ n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions become singular at all points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, self-adjoint first-order differential operators and self-adjoint elliptic pseudodifferential operators. As consequences, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a purely analytical proof of a well-known result of Branson–Gilkey [Residues of the eta function for an operator of Dirac type, J. Funct. Anal. 108(1) (1992) 47–87], which was obtained by invoking Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of [R. Ponge, Spectral asymmetry, zeta functions and the noncommutative residue, Int. J. Math. 17 (2006) 1065–1090]. Corrections to that statement are given in this paper.

Overlapping Schwarz preconditioned eigensolvers for spectral element discretizations

Model generalized eigenproblems associated with self-adjoint differential operators in nonstandard homogeneous or heterogeneous domains are considered. Their numerical approximation is based on Gauss–Lobatto–Legendre conforming spectral elements defined by Gordon–Hall transfinite mappings. The resulting discrete eigenproblems are solved iteratively with a Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method, accelerated by an overlapping Schwarz preconditioner. Several numerical tests show the good convergence properties of the proposed preconditioned eigensolver, such as its scalability and quasi-optimality in the discretization parameters, which are analogous to those obtained for overlapping Schwarz preconditioners for linear systems.

Analysis of joint spectral multipliers on Lie groups of polynomial growth

We study the problem of L p -boundedness (1<p<∞) of operators of the form m(L 1 ,⋯,L n ) for a commuting system of self-adjoint left-invariant differential operators L 1 ,⋯,L n on a Lie group G of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when G is a homogeneous group and L 1 ,⋯,L n are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.

Orthogonal polynomials associated to a certain fourth order differential equation

We introduce orthogonal polynomials \(M_{j}^{\mu,\ell}(x)\) as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters μ∈ℂ and ℓ∈ℕ0.These polynomials arise as K-finite vectors in the L 2-model of the minimal unitary representations of indefinite orthogonal groups, and reduce to the classical Laguerre polynomials \(L_{j}^{\mu}(x)\) for ℓ=0.We establish various recurrence relations and integral representations for our polynomials, as well as a closed formula for the L 2-norm. Further we show that they are uniquely determined as polynomial eigenfunctions.

On the spectrum of a two-dimensional periodic operator with a small localized perturbation

We consider a two-dimensional periodic self-adjoint second-order differential operator on the plane with a small localized perturbation. The perturbation is given by an arbitrary (not necessarily symmetric) operator. It is localized in the sense that it acts on a pair of weighted Sobolev spaces and sends functions of sufficiently rapid growth to functions of sufficiently rapid decay. By studying the spectrum of the perturbed operator, we establish that the essential spectrum is stable, the residual spectrum is absent, and the set of isolated eigenvalues is discrete. We obtain necessary and sufficient conditions for the existence of new eigenvalues arising from the ends of lacunae in the essential spectrum. In the case when such eigenvalues exist, we construct the first terms of asymptotic expansions of these eigenvalues and the corresponding eigenfunctions.

An improved parametric formulation for the variationally correct distortion immune three-noded bar element

A new method of formulation of a class of elements that are immune to mesh distortion effects is proposed here. The simple three-noded bar element with an offset of the internal node from the element center is employed here to demonstrate the method and the principles on which it is founded upon. Using the function space approach, the modified formulation is shown here to be superior to the conventional isoparametric version of the element since it satisfies the completeness requirement as the metric formulation, and yet it is in agreement with the best-fit paradigm in both the metric and the parametric domains. Furthermore, the element error is limited to only those that are permissible by the classical projection theorem of strains and stresses. Unlike its conventional counterpart, the modified element is thus not prone to any errors from mesh distortion. The element formulation is symmetric and thus satisfies the requirement of the conservative nature of problems associated with all self-adjoint differential operators. The present paper indicates that a proper mapping set for distortion immune elements constitutes geometric and displacement interpolations through parametric and metric shape functions respectively, with the metric components in the displacement/strain replaced by the equivalent geometric interpolation in parametric co-ordinates.

A stable TTI reverse time migration and its implementation

Modeling and reverse time migration based on the tilted transverse isotropic (TTI) acoustic wave equation suffers from instability in media of general inhomogeniety, especially in areas where the tilt abruptly changes. We develop a stable TTI acoustic wave equation implementation based on the original elastic anisotropic wave equation. We, specifically, derive a vertical transversely isotropic wave system of equations that is equivalent to their elastic counterpart and introduce the self-adjoint differential operators in rotated coordinates to stabilize the TTI acoustic wave equations. Compared to the conventional formulations, the new system of equations does not add numerical complexity; a stable solution can be found by either a pseudospectral method or a high-order explicit finite difference scheme. We demonstrate by examples that our method provides stable and high-quality TTI reverse time migration images.

Characterization of self-adjoint ordinary differential operators

Symmetric differential expressions ℓ of order n = 2 k with real valued coefficients give rise to self adjoint operators in the space of weighted square integrable functions. Characterization theorems exist in the literature that describe such self-adjoint operators. All such characterizations begin by constructing the maximal domain of definition of the expression ℓ . The Glazman–Krein–Naimark theorem constructs the maximal domain in terms of eigenfunctions corresponding to a nonreal parameter λ . Representations in terms of certain functions related to a real parameter λ can also be found in the literature. In this paper we construct the maximal domain from two complementary self-adjoint realizations of ℓ . One operator is assumed to be known and the other one is computed explicitly. From these two operators we explicitly give all other self-adjoint operators associated with ℓ . A special class of operators associated with ℓ is what we call Type I operators. They arise in connection with a certain bilinear form that results from the weak formulation of the expression ℓ . Depending on the deficiency index of ℓ and the properties of the bilinear form we can have two complementary self-adjoint operators (two Type I operators) and, as it turns out, one of them is the celebrated Friedrich Extension. The other operator appears to be new. As in the general case, using these two operators we give an explicit characterization of all other operators of the same Type I.

Multiplicative matrix-valued functionals and the continuity properties of semigroups corresponding to partial differential operators with matrix-valued coefficients

We define and examine certain matrix-valued multiplicative functionals with local Kato potential terms and use probabilistic techniques to prove that the semigroups of the corresponding self-adjoint partial differential operators with matrix-valued coefficients map from L 2 ( R n , C d ) to the space of continuous bounded functions, and that these semigroups have a jointly continuous and spatially bounded integral kernel. These partial differential operators include Yang–Mills type Hamiltonians with “electrical” potentials that are elements of the matrix-valued local Kato class.