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

We consider harmonic Toeplitz operators T V = P V : H(Ω) → H(Ω) where P : L 2 (Ω) → H(Ω) is the orthogonal projection onto H(Ω) = u ∈ L 2 (Ω) | ∆u = 0 in Ω , Ω ⊂ R d , d ≥ 2, is a bounded domain with boundary ∂Ω ∈ C ∞ , and V : Ω → C is an appropriate multiplier. First, we complement the known criteria which guarantee that T V is in the pth Schatten-von Neumann class S p , by simple sufficient conditions which imply T V ∈ S p,w , the weak counterpart of S p . Next, we consider symbols V ≥ 0 which have a regular power-like decay of rate γ > 0 at ∂Ω, and we show that T V is unitarily equivalent to a pseudo-differential operator of order −γ, self-adjoint in L 2 (∂Ω). Utilizing this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for T V , and establish a sharp remainder estimate. Further, we assume that Ω is the unit ball in R d , and V = V is compactly supported in Ω, and investigate the eigenvalue asymptotics of the Toeplitz operator T V . Finally, we introduce the Krein Laplacian K, self-adjoint in L 2 (Ω), perturb it by a multiplier V ∈ C(Ω; R), and show that σ ess (K + V ) = V (∂Ω). Assuming that V ≥ 0 and V |∂Ω = 0, we study the asymptotic distribution of the discrete spectrum of K ± V near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator T V .

show abstract

“…The Krein Laplacian K arises naturally in the so called abstract buckling problem (see e.g. [22,5]).…”

Section: Applications To the Spectral Theory Of The Perturbed Krein Lmentioning

confidence: 99%

Spectral properties of harmonic Toeplitz operators and applications to the perturbed Krein Laplacian

Bruneau

Raikov

2018

Asymptotic Analysis

View full text Add to dashboard Cite

show abstract

“…In this preparatory section we recall the basic facts on the Krein-von Neumann extension of a strictly positive operator S in a complex, separable Hilbert space H and its associated abstract buckling problem as discussed in [5,6]. For an extensive survey of this circle of ideas and an exhaustive list of references as well as pertinent historical comments we refer to [7].…”

Section: Basic Facts On the Krein-von Neumann Extension And The Assocmentioning

confidence: 99%

“…In particular, a nonnegative self-adjoint operator S in H is a self-adjoint extension of S if and only if S satisfies S K S S F (1.2) (again, in the sense of quadratic forms). An abstract version of [44,Proposition 1], presented in [6], describing the following intimate connection between the nonzero eigenvalues of S K , and a suitable abstract buckling problem, can be summarized as follows:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A bound for the eigenvalue counting function for Krein–von Neumann and Friedrichs extensions

Ashbaugh

Gesztesy

Лаптев

et al. 2017

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

Dedicated with great pleasure to Yuri Latushkin on the occasion of his 60th birthday. Abstract.For an arbitrary open, nonempty, bounded set Ω ⊂ R n , n ∈ N, and sufficiently smooth coefficients a, b, q, we consider the closed, strictly positive, higher-order differential operator A Ω,2m (a, b, q) in L 2 (Ω) defined on W 2m,2 0 (Ω), associated with the higher-order differential expressionand its Krein-von Neumann extension A K,Ω,2m (a, b, q) in L 2 (Ω). Denoting by N (λ; A K,Ω,2m (a, b, q)), λ > 0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of A K,Ω,2m (a, b, q), we derive the boundwhere C = C(a, b, q, Ω) > 0 (with C(In, 0, 0, Ω) = |Ω|) is connected to the eigenfunction expansion of the self-adjoint operator A 2m (a, b, q) in L 2 (R n ) defined on W 2m,2 (R n ), corresponding to τ 2m (a, b, q). Here vn := π n/2 /Γ((n + 2)/2) denotes the (Euclidean) volume of the unit ball in R n .Our method of proof relies on variational considerations exploiting the fundamental link between the Krein-von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of A 2 (a, b, q) in L 2 (R n ).We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension A F,Ω,2m (a, b, q) in L 2 (Ω) of A Ω,2m (a, b, q).No assumptions on the boundary ∂Ω of Ω are made. Date: July 4, 2018. 2010 Mathematics Subject Classification. Primary 35J25, 35J40, 35P15; Secondary 35P05, 46E35, 47A10, 47F05.Key words and phrases. Krein and Friedrichs extensions of general second-order uniformly elliptic partial differential operators, bounds on eigenvalue counting functions, spectral analysis, buckling problem.

show abstract

“…In [5] 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 ≥ εI H for some ε > 0 in a Hilbert space H to an abstract buckling problem operator is proved.…”

mentioning

confidence: 99%

Krein-von Neumann extension of an even order differential operator on a finite interval

Granovskyi¹,

Оридорога²

2018

OpMath

View full text Add to dashboard Cite

show abstract

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

Cited by 32 publications

References 24 publications

Spectral properties of harmonic Toeplitz operators and applications to the perturbed Krein Laplacian

Spectral properties of harmonic Toeplitz operators and applications to the perturbed Krein Laplacian

A bound for the eigenvalue counting function for Krein–von Neumann and Friedrichs extensions

Krein-von Neumann extension of an even order differential operator on a finite interval

Contact Info

Product

Resources

About