The von Neumann Problem for Nonnegative Symmetric Operators

Arlinskii̇̆, Yury; Tsekanovskiı̆, Eduard

doi:10.1007/s00020-003-1260-x

Cited by 39 publications

(38 citation statements)

References 29 publications

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“…The next statement provides equivalent transversalness conditions of Friedrichs and Kreȋn extensions (see [10,24]). …”

Section: Uniqueness and Transversalness Letmentioning

confidence: 95%

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Operators in divergence form and their Friedrichs and Krein extensions

Arlinskii̇̆¹,

Kovalev²

2011

OpMath

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Abstract. For a densely defined nonnegative symmetric operator A = L * 2 L1 in a Hilbert space, constructed from a pair L1 ⊂ L2 of closed operators, we give expressions for the Friedrichs and Kreȋn nonnegative selfadjoint extensions. Some conditions for the equality (L * 2 L1) * = L * 1 L2 are obtained. Applications to 1D nonnegative Hamiltonians, corresponding to point interactions, are given.

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“…The next statement provides equivalent transversalness conditions of Friedrichs and Kreȋn extensions (see [10,24]). …”

Section: Uniqueness and Transversalness Letmentioning

confidence: 95%

“…One more intrinsic construction of the Kreȋn extension A K by means of the Friedrichs extension A F has been proposed in [9] and [10]. Another approach to nonnegative selfadjoint extensions is connected with boundary triplets (boundary value spaces) and corresponding Weyl functions [3, 12-14, 18, 23].…”

Section: Introductionmentioning

confidence: 99%

Operators in divergence form and their Friedrichs and Krein extensions

Arlinskii̇̆¹,

Kovalev²

2011

OpMath

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“…7 We emphasize that the smoothness hypotheses on ∂Ω can be relaxed in the special case of the second-order Schrödinger operator associated with the differential expression −Δ + V , where V ∈ L ∞ (Ω; d n x) is real-valued: Following the treatment of self-adjoint extensions of S = (−Δ + V )| C ∞ 0 (Ω) on quasi-convex domains Ω first introduced in [19], the case of the Krein-von Neumann extension S K of S on such quasiconvex domains (which are close to minimally smooth) is treated in great detail in [9]. In particular, a Weyl-type asymptotics of the associated (nonzero) eigenvalues of S K has been proven in [9].…”

Section: −1mentioning

confidence: 97%

The Krein–von Neumann extension and its connection to an abstract buckling problem

Ashbaugh

Gesztesy²,

Mitrea

et al. 2010

Mathematische Nachrichten

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We prove the unitary equivalence of the inverse of the Krein-von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, S ≥ εIH for some ε > 0 in a Hilbert space H to an abstract buckling problem operator.In the concrete case wheren an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein-von Neumann extension of S),is in one-to-one correspondence with the problem of the buckling of a clamped plate,where u and v are related via the pair of formulaswith SF the Friedrichs extension of S. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.).

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“…We emphasize that V is not assumed to be real-valued in the bulk of this paper. One notices that 6) and, on the other hand,…”

Section: ((0 R); Dx)mentioning

confidence: 99%

Boundary Data Maps for Schrödinger Operators on a Compact Interval

Clark

Gesztesy

Mitrea

2010

Math. Model. Nat. Phenom.

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Abstract. We provide a systematic study of boundary data maps, that is, 2 × 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrödinger operators on a compact interval [0, R] with separated boundary conditions at 0 and R. Most of our results are formulated in the non-self-adjoint context.Our principal results include explicit representations of these boundary data maps in terms of the resolvent of the underlying Schrödinger operator and the associated boundary trace maps, Krein-type resolvent formulas relating Schrödinger operators corresponding to different (separated) boundary conditions, and a derivation of the Herglotz property of boundary data maps (up to right multiplication by an appropriate diagonal matrix) in the special self-adjoint case.

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The von Neumann Problem for Nonnegative Symmetric Operators

Cited by 39 publications

References 29 publications

Operators in divergence form and their Friedrichs and Krein extensions

Operators in divergence form and their Friedrichs and Krein extensions

The Krein–von Neumann extension and its connection to an abstract buckling problem

Boundary Data Maps for Schrödinger Operators on a Compact Interval

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