Mark Malamud

Generalized resolvents and the boundary value problems for Hermitian operators with gaps

A Hermitian operator A with gaps (αj, βj) (1 ⩽ j ⩽ m ⩽ ∞) is studied. The self-adjoint extensions which put exactly kj < ∞ eigenvalues into each gap (αj, βj), in particular (for kj = 0, 1 ⩽ j ⩽ m) the extensions preserving the gaps, are described in terms of boundary conditions. The generalized resolvents of the extensions with the indicated properties are described also. A solvability criterion and description of all the solutions of the Hamburger moment problem with supports in R/⋃j=1m(αj,βj) are obtained in terms of the Nevanlinna matrix.

Boundary relations and their Weyl families

The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space h, let H be an auxiliary Hilbert space, let [GRAPHICS] and let JH be defined analogously. A unitary relation G from the Krein space (h(2), J(h)) to the Kre. in space (H-2, J(H)) is called a boundary relation for the adjoint S* if ker Gamma = S. The corresponding Weyl family M(lambda) is de. ned as the family of images of the defect subspaces (n) over cap (lambda), lambda is an element of C \ R under Gamma. Here Gamma need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space H and the class of unitary relations Gamma : ( H-2, J(H)) -> (H-2, J(H)), it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every H-valued maximal dissipative (for lambda is an element of C+) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.

On the deficiency indices and self-adjointness of symmetric Hamiltonian systems

Abstract The main purpose of this paper is to investigate the formal deficiency indices N ± of a symmetric first-order system Jf′+Bf=λ H f on an interval I, where I= R or I= R ± . Here J,B, H are n×n matrix-valued functions and the Hamiltonian H ⩾0 may be singular even everywhere. We obtain two results for such a system to have minimal numbers ( N ± =0 if I= R resp. N ± =n if I= R + ) and a criterion for their maximality N ± =2n for I= R + (as well as the quasi-regularity). This covers the Kac–Krein and de Branges (Trans. Amer. Math. Soc. 99 (1961) 118) theorems on 2×2 canonical systems and some results from Kogan and Rofe–Beketov (Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75) 5). Some conditions for a canonical system to have intermediate formal deficiency indices are presented, too. We also obtain a generalization of the well known Titchmarsh–Sears theorem for second-order Sturm–Liouville-type equations. This contains results due to Lidskii and Krein as special cases. We present two approaches to the above problems: one dealing with formal deficiency indices and one dealing with (ordinary) deficiency indices. Our main (non-formal) approach is based on the investigation of a symmetric linear relation Smin which is naturally associated to a first-order system. This approach works in the framework of extension theory and therefore we investigate in detail the domain D (S min ∗ ) of S min ∗ . In particular, we prove the so called regularity theorem for D (S min ∗ ) . As a byproduct of the regularity result we obtain very short proofs of (generalizations of) the main results of the paper by Kogan and Rofe–Beketov (1974/75).

Weyl function and spectral properties of self-adjoint extensions

We characterize the spectra of self-adjoint extensions of a symmetric operator with equal deficiency indices in terms of boundary values of their Weyl functions. A complete description is obtained for the point and absolutely continuous spectrum while for the singular continuous spectrum additional assumptions are needed. The results are illustrated by examples.

Boundary relations and generalized resolvents of symmetric operators

The Kreĭn-Naĭmark formula provides a parametrization of all selfadjoint exit space extensions of a (not necessarily densely defined) symmetric operator in terms of maximal dissipative (in ℂ+) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct generalized resolvents from given parameter families. A general version of the coupling method is introduced and the role of the boundary relations and their Weyl families in the Kreĭn-Naĭmark formula is investigated and explained. These notions lead to several new results and new types of solutions to problems involving generalized resolvents and their applications, e.g., in boundary-value problems for (ordinary and partial) differential operators. For instance, an old problem going back to M. A. Naĭmark and concerning the analytic characterization of the so-called Naĭmark extensions is solved.

Scattering matrices and Weyl functions

For a scattering system {A Θ , A 0 } consisting of self-adjoint extensions A Θ and A 0 of a symmetric operator A with finite deficiency indices, the scattering matrix {S Θ (λ)} and a spectral shift function ξ Θ are calculated in terms of the Weyl function associated with a boundary triplet for A*, and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar and matrix potentials, to Dirac operators and to Schrodinger operators with point interactions.

SPECTRAL THEORY OF ELLIPTIC OPERATORS IN EXTERIOR DOMAINS

AbstractDiverse closed (and selfadjoint) realizations of elliptic differential expressions A = Σ0⩽|α|,|β|⩽m(−1)αDαaα,β(x)Dβ, aα,β(·) ∈ C∞(

The Similarity Problem for J-nonnegative Sturm-Liouville Operators

\bar \Omega