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Book ChapterDOI

M. Kreĭn’s Research on Semi-Bounded Operators, its Contemporary Developments, and Applications

01 Jan 2009-pp 65-112
TL;DR: In this paper, the authors consider the M Kreĭn classical papers on semi-bounded operators and the theory of contractive self-adjoint extensions of Hermitian contractions, and discuss their impact and role in the solution of J von Neumann's problem about parametrization in terms of his formulas of all nonnegative selfadjoint extension of nonnegative symmetric operators.
Abstract: We are going to consider the M Kreĭn classical papers on the theory of semi-bounded operators and the theory of contractive self-adjoint extensions of Hermitian contractions, and discuss their impact and role in the solution of J von Neumann’s problem about parametrization in terms of his formulas of all nonnegative self-adjoint extensions of nonnegative symmetric operators, in the solution of the Phillips-Kato extension problems (in restricted sense) about existence and parametrization of all proper sectorial (accretive) extensions of nonnegative operators, in bi-extension theory of non-negative operators with the exit into triplets of Hilbert spaces, in the theory of singular perturbations of nonnegative self-adjoint operators, in general realization problems (in system theory) of Stieltjes matrix-valued functions, in Nevanlinna-Pick system interpolation in the class of sectorial Stieltjes functions, in conservative systems theory with accretive main Schrodinger operator, in the theory of semi-bounded symmetric and self-adjoint operators invariant with respect to some groups of transformations New developments and applications to the singular differential operators are discussed as well
Citations
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Book
01 Jan 1978

154 citations

Journal ArticleDOI
TL;DR: In this article, a description of all self-adjoint extensions of the Laplacian in quasiconvex domains is given, where the domain Ω belongs to a subclass of bounded Lipschitz domains (which are termed quasi-convex) and all convex domains as well as all domains of class C ≥ 1/2.
Abstract: This paper has two main goals. First, we are concerned with a description of all self-adjoint extensions of the Laplacian $$ - \Delta {|_{C_0^\infty (\Omega )}}$$ in L 2(Ω; d n x). Here, the domain Ω belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), that contains all convex domains as well as all domains of class C 1,r , for r > 1/2. Second, we establish Kreĭn-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasiconvex domains and study the well-posedness of boundary value problems for the Laplacian as well as basic properties of the corresponding Weyl-Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps). One significant innovation in this paper is an extension of the classical boundary trace theory for functions in spaces that lack Sobolev regularity in a traditional sense, but are suitably adapted to the Laplacian.

121 citations

Journal ArticleDOI
TL;DR: In this paper, the authors systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals (a, b) associated with rather general differential expressions of the type \[ ======\tau f = \frac{1}{r} (- \big(p[f' + s f]\big)' + s p[f+s f] + qf),] where the coefficients $p, $q, $r, $s$ are real-valued and Lebesgue measurable on $(a,b)$, with $
Abstract: We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[ \tau f = \frac{1}{r} (- \big(p[f' + s f]\big)' + s p[f' + s f] + qf),] where the coefficients $p$, $q$, $r$, $s$ are real-valued and Lebesgue measurable on $(a,b)$, with $p eq 0$, $r>0$ a.e.\ on $(a,b)$, and $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$, and $f$ is supposed to satisfy [f \in AC_{\text{loc}}((a,b)), \; p[f' + s f] \in AC_{\text{loc}}((a,b)).] In particular, this setup implies that $\tau$ permits a distributional potential coefficient, including potentials in $H^{-1}_{\text{loc}}((a,b))$. We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator $T_{\text{max}}$, or equivalently, all self-adjoint extensions of the minimal operator $T_{\text{min}}$, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of $T_{\text{min}}$. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira $m$-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of $T_{\text{min}}$. Finally, in the special case where $\tau$ is regular, we characterize the Krein-von Neumann extension of $T_{\text{min}}$ and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).

88 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the perturbed Krein Laplacian (i.e., the Krein-von Neumann extension of − Δ + V defined on C 0 ∞ ( Ω ) is spectrally equivalent to the buckling of a clamped plate problem.

78 citations

Journal ArticleDOI
TL;DR: In this paper, the authors systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum.
Abstract: We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals \((a,b) \subseteq \mathbb{R}\) associated with rather general differential expressions of the type \begin{equation*}\tau f = \frac{1}{\tau} (-(p[f'+sf])'+sp[f'+sf]+qf),\end{equation*} where the coefficients \(p, q, r, s\) are real-valued and Lebesgue measurable on \((a,b)\), with \(p eq 0\), \(r \gt 0\) a.e. on \((a,b)\), and \(p^{-1}, q, r, s \in L_{loc}^1((a,b),dx)\), and \(f\) is supposed to satisfy \begin{equation*} f \in AC_{loc}((a,b)), p[f'+sf] \in AC_{loc}((a,b)). \end{equation*} In particular, this setup implies that \(\tau\) permits a distributional potential coefficient, including potentials in \(H_{loc}^{-1}((a,b))\). We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator \(T_{max}\), or equivalently, all self-adjoint extensions of the minimal operator \(T_{min}\), all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of \(T_{min}\). In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira m-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of \(T_{min}\). Finally, in the special case where \(\tau\) is regular, we characterize the Krein-von Neumann extension of \(T_{min}\) and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).

73 citations

References
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Book
01 Jan 1966
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Abstract: "The monograph by T Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4) Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8) The fundamentals of semigroup theory are given in chapter 9 The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10 The first edition is now 30 years old The revised edition is 20 years old Nevertheless it is a standard textbook for the theory of linear operators It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field Zentralblatt MATH, 836

19,846 citations

Book
01 Jan 1970
TL;DR: In this article, the structure of operators of Class C 0.1 is discussed.Contractions and their Dilations, Geometrical and Spectral Properties of Dilations and Operator-Valued Analytic Functions are discussed.
Abstract: Contractions and Their Dilations.- Geometrical and Spectral Properties of Dilations.- Functional Calculus.- Extended Functional Calculus.- Operator-Valued Analytic Functions.- Functional Models.- Regular Factorizations and Invariant Subspaces.- Weak Contractions.- The Structure of C1.-Contractions.- The Structure of Operators of Class C0.

2,339 citations

Book
01 Jan 1988
TL;DR: The one-center point interaction as discussed by the authors is a special case of the Coulomb point interaction, where Coulomb plus one center point interaction in three dimensions plus Coulomb and one center interaction in two dimensions.
Abstract: Introduction The one-center point interaction: The one-center point interaction in three dimensions Coulomb plus one-center point interaction in three dimensions The one-center $\delta$-interaction in one dimension The one-center $\delta$'-interaction in one dimension The one-center point interaction in two dimensions Point interactions with a finite number of centers: Finitely many point interactions in three dimensions Finitely many $\delta$-interactions in one dimension Finitely many $\delta$'-interactions in one dimension Finitely many point interactions in two dimensions Point interactions with infinitely many centers: Infinitely many point interactions in three dimensions Infinitely many $\delta$-interactions in one dimension Infinitely many $\delta$'-interactions in one dimension Infinitely many point interactions in two dimensions Random Hamiltonians with point interactions Appendices: Self-adjoint extensions of symmetric operators Spectral properties of Hamiltonians defined as quadratic forms Schrodinger operators with interactions concentrated around infinitely many centers Boundary conditions for Schrodinger operators on $(0,\infty)$ Time-dependent scattering theory for point interactions Dirichlet forms for point interactions Point interactions and scales of Hilbert spaces Nonstandard analysis and point interactions Elements of probability theory Relativistic point interactions in one dimension References Author Index Subject Index Seize ans apres Bibliography Errata and addenda.

1,806 citations

Journal ArticleDOI
TL;DR: In this paper, a Hermitian operator A with gaps (αj, βj) (1 ⩽ j⩽ m ⩾ ∞) is studied and the self-adjoint extensions which put exactly kj < ∞ eigenvalues into each gap are described in terms of boundary conditions.

598 citations