Left-definite theory with applications to orthogonal polynomials

In the past several years, there has been considerable progress made on a general left-definite theory associated with a self-adjoint operator A that is bounded below in a Hilbert space H; the term ‘left-definite’ has its origins in differential equations but Littlejohn and Wellman [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280-339] generalized the main ideas to a general abstract setting. In particular, it is known that such an operator A generates a continuum {H_r}_r>0 of Hilbert spaces and a continuum of {A_r}_r>0 of self-adjoint operators. In this paper, we review the main theoretical results in [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280-339]; moreover, we apply these results to several specific examples, including the classical orthogonal polynomials of Laguerre, Hermite, and Jacobi.     

Further study of entire radial solutions of a biharmonic equation with exponential nonlinearity

We study the structure of entire radial solutions of a biharmonic equation with exponential nonlinearity: $$\begin{array}{ll}\Delta^2 u = \lambda {\rm e}^u \;\; {\rm in}\; \mathbb{R}^N, N \geq 5 \quad\quad\quad (0.1)\end{array}$$ with λ = 8(N ? 2)(N ? 4). It is known from a recent interesting paper by Arioli et al. that (0.1) admits a singular solution U _s (r) = ln r ^?4. We show that for 5 ≤ N ≤ 12, any regular entire radial solution u with u(r) ? ln r ^?4 → 0 as r → ∞ of (0.1) intersects with U _s (r) infinitely many times. On the other hand, if N ≥ 13, then u(r) < U _s (r) for all r > 0, and the solutions are strictly ordered with respect to the initial value a = u(0). Moreover, the asymptotic expansions of the entire radial solutions near ∞ are also obtained. Our main results give a positive answer to a conjecture in Arioli et al. (J Differ Equ 230:743–770, 2006) [see lines ?11 to ?9, p. 747 of Arioli et al. (J Differ Equ 230:743–770, 2006)].     

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.     

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).     

Spectral shift function of higher order

This paper resolves affirmatively Koplienko’s (Sib. Mat. Zh. 25:62–71, 1984) conjecture on existence of higher order spectral shift measures. Moreover, the paper establishes absolute continuity of these measures and, thus, existence of the higher order spectral shift functions. A spectral shift function of order n∈? is the function η _n =η _n,H,V such that $$ \operatorname {Tr}\Biggl( f(H + V)-\sum_{k = 0}^{n-1} \frac{1}{k!}\, \frac{d^k}{dt^k} \bigl[ f(H + tV) \bigr] \bigg|_{t = 0} \Biggr) = \int_\mathbb{R}f^{(n)} (t)\, \eta_n (t)\, dt, $$ for every sufficiently smooth function f, where H is a self-adjoint operator defined in a separable Hilbert space ? and V is a self-adjoint operator in the n-th Schatten-von Neumann ideal S ⁿ . Existence and summability of η ₁ and η ₂ were established by Krein (Mat. Sb. 33:597–626, 1953) and Koplienko (Sib. Mat. Zh. 25:62–71, 1984), respectively, whereas for n>2 the problem was unresolved. We show that η _n,H,V exists, integrable, and $$\Vert \eta_n \Vert _{L^1(\mathbb{R})} \leq c_n \Vert V \Vert _{S^n}^n, $$ for some constant c _n depending only on n∈?. Our results for η _n rely on estimates for multiple operator integrals obtained in this paper. Our method also applies to the general semi-finite von Neumann algebra setting of the perturbation theory.     

Commutators, C0-semigroups and resolvent estimates

We study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self-adjoint operator H in the framework of the conjugate operator method initiated by Mourre. We allow the conjugate operator A to be the generator of a C₀-semigroup (finer estimates require A to be maximal symmetric) and we consider situations where the first commutator [H,iA] is not comparable to H. The applications include the spectral theory of zero mass quantum field models.     

Invariant Subspaces for Commuting Operators on a Real Banach Space

It is proved that the commutative algebra A of operators on a reflexive real Banach space has an invariant subspace if each operator T ∈ A satisfies the condition

$${\left\| {1 - \varepsilon {T^2}} \right\|_e} \leqslant 1 + o\left( \varepsilon \right)as\varepsilon \searrow 0,$$

where ║ · ║ _e denotes the essential norm. This implies the existence of an invariant subspace for any commutative family of essentially self-adjoint operators on a real Hilbert space.

     

The local determination of Jordan bases for H-self-adjoint operators

Let H be an invertible self-adjoint operator on a finite dimensional Hilbert space $\text{X}$. A linear operator A is said to be H-self-adjoint (or self-adjoint relative to H) if HA = A¹H. Let σ(A) denote, as usual, the spectrum of A. If A is H-self-adjoint, then A is similar to A¹ and λ ∈ σ(A) implies λ&#x0304; ∈ σ (A), so that the spectrum of A issymmetric with respect to the real axis. Given spectral information for A at an eigenvalue λ₀ (≠ λ&#x0304;₀), we investigate the corresponding information at λ&#x0304;₀ and, in particular, the unique pairing of Jordan bases for the root subspaces at λ₀ and λ&#x0304;₀.     

The solutions of a variational inequality as critical points of a real valued smooth function

Let H be a real Hilbert space, A : H → H a self-adjoint continuous linear operator. Given a variational inequality related to A, we construct a Lipschitz continuously differentiable function F:H→$\text{R}$ so that the critical points of F are the solutions of the variational inequality.     

Generalized sum of operators

For a semibounded below self-adjoint operatorA in a Hilbert spaceH and a singular operatorV acting in theA-scale of Hilbert spaces, the notion of generalized sumA?V is introduced. Conditions are found forA?V to be self-adjoint in ?. In particular, it is shown that if a symmetric operatorV is semibounded or has a spectral gap, then there exists an α such that the generalized sumA?αV is a self-adjoint operator inH. For a symmetric restrictionA = A‖D, D C D(A), with deficiency indices (1, 1), it is proved that each self-adjoint extension à of A admits representation as a generalized sum Ã=A?V.     

Regularity for solutions of non local parabolic equations

We study the regularity of solutions of parabolic fully nonlinear nonlocal equations. We prove C ^α regularity in space and time and, under different assumptions on the kernels, C ^1,α in space for translation invariant equations. The proofs rely on a weak parabolic ABP and the classic ideas of Tso (Commun. Partial Diff. Equ. 10(5):543–553, 1985) and Wang (Commun. Pure Appl. Math. 45(1), 27–76, 1992). Our results remain uniform as σ → 2 allowing us to recover most of the regularity results found in Tso (Commun. Partial Diff. Equ. 10(5):543–553, 1985).     

Unconditional uniqueness in the charge class for the Dirac–Klein–Gordon equations in two space dimensions

Recently, Grünrock and Pecher proved global well-posedness of the 2d Dirac–Klein–Gordon equations given initial data for the spinor and scalar fields in H ^s and H ^s+1/2 × H ^s-1/2, respectively, where s ≥ 0, but uniqueness was only known in a contraction space of Bourgain type, strictly smaller than the natural solution space C([0,T]; H ^s × H ^s+1/2 × H ^s-1/2). Here we prove uniqueness in the latter space for s ≥ 0. This improves a recent result of Pecher, where the range s > 1/30 was covered.     

A Resolvent Approach to the Real Quantum Plane

Let q ≠ ± 1 be a complex number of modulus one. This paper deals with the operator relation AB = qBA for self-adjoint operators A and B on a Hilbert space. Two classes of well-behaved representations of this relation are studied in detail and characterized by resolvent equations.     

Ranges of nonlinear asymptotically linear operators

This paper deals with the solvability of the equation A(u) ? S(u) = f, where A is a continuous self-adjoint operator defined on a real Hilbert space H with values in H, the null-space of A is nontrivial, and N is a nonlinear completely continuous perturbation. Sufficient, and necessary-sufficient conditions are given for the equation to be solvable. Abstract theorems are applied to solving boundary value problems for partial differential equations.     

Singularly perturbed self-adjoint operators in scales of Hilbert spaces

Finite-rank perturbations of a semibounded self-adjoint operator A are studied in a scale of Hilbert spaces associated with A. The notion of quasispace of boundary values is used to describe self-adjoint operator realizations of regular and singular perturbations of the operator A by the same formula. As an application, the one-dimensional Schrödinger operator with generalized zero-range potential is studied in the Sobolev space W ₂ ^p (?), p ∈ ?.     

On symmetries in the theory of finite rank singular perturbations

For a nonnegative self-adjoint operator A₀ acting on a Hilbert space H singular perturbations of the form A₀+V, are studied under some additional requirements of symmetry imposed on the initial operator A₀ and the singular elements ψj. A concept of symmetry is defined by means of a one-parameter family of unitary operators U that is motivated by results due to R.S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials V and the corresponding self-adjoint realizations of A₀+V. The results are applied for the investigation of singular perturbations of the Schrödinger operator in L₂(R³) and for the study of a (fractional) p-adic Schrödinger type operator with point interactions.     

Quasi-Reversibility Method for an Ill-Posed Nonhomogeneous Parabolic Problem

The quasi-reversibility method is considered for the non-homogeneous backward Cauchy problem u_t+Au = f(t), u(τ) = ? for 0≤t<τ, which is known to be an ill-posed problem. Here, A is a densely defined positive self-adjoint unbounded operator on a Hilbert space H with given data f∈L¹([0,τ],H) and ?∈H. Error analysis is considered when the data ?, f are exact and also when they are noisy. The results obtained generalize and simplify many of the results available in the literature.     

Dissipators for symmetric quasi-free dynamical semigroups on the CAR algebra

Let L be a negative self-adjoint bounded operator on a Hilbert space $\text{H}$, and p a projection on $\text{H}$ with pLp trace class, and let {T_t: t ? 0} be the extension of {e^tL: t ? 0} to a strongly continuous semigroup of completely positive quasi-free unital maps of Fock type on the fermion algebra A$\text{H}$ built over $\text{H}$. Then it is shown that there exists a strongly continuous self-adjoint contraction semigroup {G_t: t ? 0} on the Hilbert space of the GNS decomposition of the quasi-free state gw_p such that in the representation of that state: T_t ? G_t(·)G_t, t ?0.