On the Spectra of Left-Definite Operators

If A is a self-adjoint operator that is bounded below in a Hilbert space H, Littlejohn and Wellman (J Diff Equ 181(2):280–339, 2002) showed that, for each r > 0, there exists a unique Hilbert space H _r and a unique self-adjoint operator A _r in H _r satisfying certain conditions dependent on H and A. The space H _r and the operator A _r are called, respectively, the rth left-definite space and rth left-definite operator associated with (H, A). In this paper, we show that the operators A, A _r , and A _s (r, s > 0) are isometrically isomorphically equivalent and that the spaces H, H _r , and H _s (r, s > 0) are isometrically isomorphic. These results are then used to reproduce the left-definite spaces and left-definite operators. Furthermore, we will see that our new results imply that the spectra of A and A _r are equal, giving us another proof of this phenomenon that was first established in Littlejohn and Wellman (J Diff Equ 181(2):280–339, 2002).     

Characterization of domains of self-adjoint ordinary differential operators

The GKN (Glazman, Krein, Naimark) Theorem characterizes all self-adjoint realizations of linear symmetric (formally self-adjoint) ordinary differential equations in terms of maximal domain functions. These functions depend on the coefficients and this dependence is implicit and complicated. In the regular case an explicit characterization in terms of two-point boundary conditions can be given. In the singular case when the deficiency index d is maximal the GKN characterization can be made more explicit by replacing the maximal domain functions by a solution basis for any real or complex value of the spectral parameter λ. In the much more difficult intermediate cases, not all solutions contribute to the singular self-adjoint conditions. In 1986 Sun found a representation of the self-adjoint singular conditions in terms of certain solutions for nonreal values of λ. In this paper we give a representation in terms of certain solutions for real λ. This leads to a classification of solutions as limit-point (LP) or limit-circle (LC) in analogy with the celebrated Weyl classification in the second-order case. The LC solutions contribute to the singular boundary conditions, the LP solutions do not. The advantage of using real λ is not only because it is, in general, easier to find explicit solutions but, more importantly, it yields information about the spectrum.     

Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations

This paper investigates the effects of a degenerate diffusion term in reaction-diffusion models u_t=[D(u)u_x]_x+g(u) with Fisher-KPP type g. Both in the case when D(0)=0 and when D(0)=D(1)=0, with D(u)>0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c∗ and we show the appearance of a sharp-type profile when c=c∗. These results solve recent conjectures formulated by Sánchez-Garduño and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339).     

The sum of unitary similarity orbits containing only special operators

Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A₁,…,A_k∈B(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A∈B(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={U^∗AU:Uunitary} always lie in S.     

A generalized moment problem for self-adjoint operators

Let the sequence {λ _i } (i≧0) satisfy condition (1.1) and let {A _n} (n≧0) be a sequence of bounded self-adjoint operators over a complex Hilbert spaceH. We give a necessary and sufficient condition in order that {A _n} (n≧0) should possess the representation (1.2).     

Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A,B in the self-adjoint Jacobi operator H=AS⁺+A^-S^-+B (with S^± the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E₊], E_-<E₊, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D with spectrum given by , 0?E_-<E₊.Our approach is based on trace formulas and matrix-valued (exponential) Herglotz representation theorems. As a by-product of our techniques we obtain the extension of Flaschka's Borg-type result for periodic scalar Jacobi operators to the class of reflectionless matrix-valued Jacobi operators.     

Quantum effects, sequential independence and majorization

A quantum effect is a positive Hilbert space contraction operator. If {E_i}, 1?i?n, are n quantum effects (defined on some Hilbert space H), then their sequential product is the operator . It is proved that the quantum effects {E_i}, 1?i?n, are sequentially independent if and only if for every permutation r₁r₂…r_n of the set S_n={1,2,…,n}. The sequential independence of the effects E_i, 1?i?n, implies E_noE_n-1o…oE_j+1oE_jo…oE₁=(E_noE_n-1o…E_j+1)oE_jo…oE₁ for every 1?j?n. It is proved that if there exists an effect E_j, 1?j?n, such that E_j?(E_noE_n-1o…E_j+1)oE_jo…oE₁, then the effects {E_i} are sequentially independent and satisfy .     

Finiteness of the discrete spectrum of some block Toeplitz operators

Sufficient conditions are given for the finiteness of the discrete spectrum of the block Toeplitz operatorT _A generated in the spaceH ₂ ⁿ by self-adjoint matrix functionA(t)(|t|=1). These results are obtained by means of theorems concerning the spectrum of a perturbed self-adjoint operators.     

Mourre's inequality and embedded bound states

A self-adjoint operator H with an eigenvalue λ embedded in the continuum spectrum is considered. Boundedness of all operators of the form AnP is proved, where P is the eigenprojection associated with λ and A is any self-adjoint operator satisfying Mourre's inequality in a neighborhood of λ and such that the higher commutators of H with A up to order n+2 are relatively bounded with respect to H.     

Strongly damped wave problems: Bootstrapping and regularity of solutions

The aim of the article is to present a unified approach to the existence, uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in Banach spaces. Our results are applied next to a number of examples appearing in literature, which fall into the class of strongly damped semilinear wave equations. The present work essentially extends the results on the existence and regularity of solutions to such problems. Previously, these problems have been considered mostly within the Hilbert space setting and with the main part operators being selfadjoint. In this article we present a more general approach, involving sectorial operators in reflexive Banach spaces.     

Schatten p-norm inequalities related to an extended operator parallelogram law

Let C_p be the Schatten p-class for p>0. Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: if A={A₁,A₂,…,A_n} and B={B₁,B₂,…,B_n} are two sets of operators in, then C₂

     

Singular values, norms, and commutators

Let and X_i, i=1,…,n, be bounded linear operators on a separable Hilbert space such that X_i is compact for i=1,…,n. It is shown that the singular values of are dominated by those of , where ‖·‖ is the usual operator norm. Among other applications of this inequality, we prove that if A and B are self-adjoint operators such that a₁?A?a₂ and b₁?B?b₂ for some real numbers and b₂, and if X is compact, then the singular values of the generalized commutator AX-XB are dominated by those of max(b₂-a₁,a₂-b₁)(X⊕X). This inequality proves a recent conjecture concerning the singular values of commutators. Several inequalities for norms of commutators are also given.     

Approximation of isolated eigenvalues of general singular ordinary differential operators

Let A be a self-adjoint operator defined by a general singular ordinary differential expression τ on an interval (a, b), ? ∞ ≤ a < b ≤ ∞. We show that isolated eigenvalues in any gap of the essential spectrum of A are exactly the limits of eigenvalues of suitably chosen self-adjoint realizations A_n of τ on subintervals (a_n, b_n) of (a, b) with a_n → a, b_n → b. This means that eigenvalues of singular ordinary differential operators can be approximated by eigenvalues of regular operators. In the course of the proof we extend a result, which is well known for quasiregular differential expressions, to the general case: If the spectrum of A is not the whole real line, then the boundary conditions needed to define A can be given using solutions of (τ ? λ)u = 0, where λ is contained in the regularity domain of the minimal operator corresponding to τ.     

Self-adjointness of relatively bounded quadratic forms and operators

We consider a Hilbert space $\text{H}$ on which is given a positive self-adjoint operator H. For densely defined bilinear forms or operators A we obtain conditions which ensure that A is an operator, that A is self-adjoint and that e^iAt leaves D(H^r) invariant with H^re^iAt strongly differentiable.     

1-D Schrödinger operators with local point interactions on a discrete set

Spectral properties of 1-D Schrödinger operators with local point interactions on a discrete set are well studied when d_∗:=infn,k∈N|xn−xk|>0. Our paper is devoted to the case d_∗=0. We consider HX,α in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions.We show that the spectral properties of HX,α like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators HX,α to be self-adjoint, lower semibounded, and discrete in the case d_∗=0.The operators with δ^′-type interactions are investigated too. The obtained results demonstrate that in the case d_∗=0, as distinguished from the case d_∗>0, the spectral properties of the operators with δ- and δ^′-type interactions are substantially different.     

Convergence estimates of a projection-difference method for an operator-differential equation

This article investigates the projection-difference method for a Cauchy problem for a linear operator-differential equation with a leading self-adjoint operator A(t) and a subordinate linear operator K(t) in Hilbert space. This method leads to the solution of a system of linear algebraic equations on each time level; moreover, the projection subspaces are linear spans of eigenvectors of an operator similar to A(t). The convergence estimates are obtained. The application of the developed method for solving the initial boundary value problem is given.     

On the Right-Definite and Left-Definite Spectral Theory of the Legendre Polynomials

In this paper, we further develop the left-definite and right-definite spectral theory associated with the self-adjoint differential operator A in L²(-1,1), generated from the classical second-order Legendre differential equation, having the sequence of Legendre polynomials as eigenfunctions. Specifically, we determine the first three left-definite spaces associated with the pair (L²(-1,1),A). As a consequence of these results, we determine the explicit domain of both the associated left-definite operator A₁, first observed by Everitt, and the self-adjoint operator A^1/2. In addition, we give a new characterization of the domain D(A) of A and, as a corollary, we present a new proof of the Everitt-Mari result which gives optimal global smoothness of functions in D(A).     

An optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems

In this paper, we modify the adaptive wavelet algorithm from Gantumur et al. [An optimal adaptive wavelet method without coarsening of the iterands, Technical Report 1325, Department of Mathematics, Utrecht University, March 2005, Math. Comp., to appear] so that it applies directly, i.e., without forming the normal equation, not only to self-adjoint elliptic operators but also to operators of the form L=A+B$L=A+B$, where A is self-adjoint elliptic and B is compact, assuming that the resulting operator equation is well posed. We show that the algorithm has optimal computational complexity.