The von Neumann Problem for Nonnegative Symmetric Operators

doi:10.1007/S00020-003-1260-X

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The von Neumann Problem for Nonnegative Symmetric Operators

Journal Article•DOI•

The von Neumann Problem for Nonnegative Symmetric Operators

Yury Arlinskii, Eduard Tsekanovskii¹•Institutions (1)

Niagara University¹

01 Mar 2005-Integral Equations and Operator Theory (Birkhäuser-Verlag)-Vol. 51, Iss: 3, pp 319-356

TL;DR: In this paper, the authors present an independent solution to von Neumann's problem on the parametrization in explicit form of all nonnegative self-adjoint extensions of a densely defined nonnegative symmetric operator.

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Abstract: We develop a new approach and present an independent solution to von Neumann’s problem on the parametrization in explicit form of all nonnegative self-adjoint extensions of a densely defined nonnegative symmetric operator. Our formulas are based on the Friedrichs extension and also provide a description for closed sesquilinear forms associated with nonnegative self-adjoint extensions. All basic results of the well-known Krein and Birman-Vishik theory and its complementations are derived as consequences from our new formulas, including the parametrization (in the framework of von Neumann’s classical formulas) for all canonical resolvents of nonnegative selfadjoint extensions. As an application all nonnegative quantum Hamiltonians corresponding to point-interactions in $\mathbb{R}^3$ are described.

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Journal Article•DOI•

A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains

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Fritz Gesztesy¹, Marius Mitrea¹•Institutions (1)

University of Missouri¹

17 Apr 2011-Journal D Analyse Mathematique

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.

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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.

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121 citations

Book•

Boundary Value Problems, Weyl Functions, and Differential Operators

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Jussi Behrndt, Seppo Hassi, Henk De Snoo

08 Oct 2020

110 citations

Cites background from "The von Neumann Problem for Nonnega..."

..., in [36, 43, 44, 47, 48, 54, 56, 57, 248, 301, 357, 359, 363, 553, 555, 556], where also accretive and sectorial extensions, and associated closed forms are treated, and see also [552, 580, 581, 582, 636, 725]....
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Posted Content•

Generalized Robin Boundary Conditions, Robin-to-Dirichlet Maps, and Krein-Type Resolvent Formulas for Schrödinger Operators on Bounded Lipschitz Domains

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Fritz Gesztesy, Marius Mitrea

21 Mar 2008-arXiv: Analysis of PDEs

TL;DR: In this article, generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrodinger operators on bounded Lipschitz domains were studied.

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Abstract: We study generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schr\"odinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$. We also discuss the case of bounded $C^{1,r}$-domains, $(1/2)

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89 citations

Additional excerpts

...84], [4], [8], [9], [12], [14], [15], [21], [22], [38], [41], [48], [50]–[57], [60], [61], [66], [72]–[79], [84], [87]–[89], and the references cited therein....
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Journal Article•DOI•

Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials

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Jonathan Eckhardt¹, Fritz Gesztesy², Roger Nichols³, Gerald Teschl¹•Institutions (3)

University of Vienna¹, University of Missouri², University of Tennessee at Chattanooga³

23 Aug 2012-arXiv: Spectral Theory

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 $

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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).

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88 citations

Journal Article•DOI•

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

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Mark S. Ashbaugh¹, Fritz Gesztesy¹, Marius Mitrea¹, Gerald Teschl²•Institutions (2)

University of Missouri¹, Schrödinger²

01 Mar 2010-Advances in Mathematics

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.

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78 citations

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References

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Book•

Perturbation theory for linear operators

[...]

Tosio Kato

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.

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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

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19,846 citations

"The von Neumann Problem for Nonnega..." refers background in this paper

...As it is well known [29] the Friedrichs extension SF of S is defined as a nonnegative self-adjoint extension associated with the form S[·, ·]....
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...Recall some definitions and results from [29]....
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...A form τ is closed if and only if the linear manifold D[τ ] is a Hilbert space with the inner product (u, v)τ = τ [u, v] + (u, v) [29]....
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...If τ is a closed, densely defined nonnegative form, then according to the first representation theorem [29] there exists a unique nonnegative self-adjoint operator T in H, associated with τ in the following sense: (Tu, v) = τ [u, v] for all u ∈ D(T ) and for all v ∈ D[τ ]....
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...By the second representation theorem [29] the identities hold: D[τ ] = D(T ), τ [u, v] = (T u, T v), u, v ∈ D[τ ]....
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Book•

Solvable Models in Quantum Mechanics

[...]

Sergio Albeverio

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.

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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.

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1,806 citations

"The von Neumann Problem for Nonnega..." refers background or methods in this paper

...We also give a description of all nonnegative Schrödinger operators corresponding to point-interactions in R [1] and find the minimal Krĕınvon Neumann nonnegative self-adjoint extension....
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...There are many publications on nonnegative self-adjoint and non-selfadjoint accretive extensions and their applications, and we refer in this matter to [1]–[53]....
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...As it is well known [1] the operator S is a nonnegative symmetric operator in L(R, dx) with defect numbers 〈 m,m 〉 and its the Friedrichs extension SF is given by D(SF ) = H 2 (R), SF = −∆....
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Journal Article•DOI•

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

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Vladimir Derkach, Mark Malamud

01 Jan 1991-Journal of Functional Analysis

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.

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598 citations

"The von Neumann Problem for Nonnega..." refers background in this paper

...As it was shown in [19], [20] the operators S0, S1 defined as follows Sk = S∗ KerΓk, k = 0, 1 are transversal to each other self-adjoint extensions of S....
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...In particular, the domains of the extremal Friedrichs and Krĕın-von Neumann extensions can be described in external terms by means of the limit values of the Weyl-Titchmarsh functions at 0 and −∞ [18], [19], [20], [21], [24], [26], [42], [48]....
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...Malamud [18], [19], [20] define the Weyl (Weyl-Titchmarsh) function M0(λ) by the equality M0(λ) = Γ1Γ0(λ)....
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...It was established [18], [19], [20] (see also [21], [24], [26], [42]) the following theorem....
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Journal Article•DOI•

Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren

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J. von Neumann

01 Dec 1930-Mathematische Annalen

504 citations

"The von Neumann Problem for Nonnega..." refers background or methods in this paper

...SN coincides with the operator constructed by J. von Neumann in [ 44 ]....
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...and the minimal ones, such that the maximal one coincides with the Friedrichs extension and the minimal one becomes the extension obtained by J. von Neumann [ 44 ] in the case of a symmetric operator with a positive lower bound (positive definite operator)....
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...In 1929 John von Neumann published a paper [ 44 ] where for the first time extension theory and his well-known formulas describing the domains (in terms of operator-valued parameter) of all self-adjoint extensions of a given symmetric operator acting on some Hilbert space appeared....
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Book•

Linear transformations in Hilbert space and their applications to analysis

[...]

M. H. Stone

01 Jan 1932

TL;DR: In this paper, the unitary equivalence of self-adjoint transformations is proved for linear transformations in Hilbert space, and the operational calculus is used to prove the equivalence.

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Abstract: Hilbert space and its realizations Transformations in Hilbert space Examples of linear transformations Resolvents, spectra, reducibility Self-adjoint transformations The operational calculus The unitary equivalence of self-adjoint transformations General types of linear transformations Applications Index.

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392 citations