## A SURVEY ON THE KREIN–VON NEUMANN EXTENSION, THE CORRESPONDING ABSTRACT BUCKLING PROBLEM, AND WEYL-TYPE SPECTRAL ASYMPTOTICS FOR PERTURBED KREIN LAPLACIANS IN NONSMOOTH DOMAINS (2012)

### BibTeX

@MISC{Ashbaugh12asurvey,

author = {Mark S. Ashbaugh and Fritz Gesztesy and Marius Mitrea and Roman Shterenberg and Gerald Teschl},

title = {A SURVEY ON THE KREIN–VON NEUMANN EXTENSION, THE CORRESPONDING ABSTRACT BUCKLING PROBLEM, AND WEYL-TYPE SPECTRAL ASYMPTOTICS FOR PERTURBED KREIN LAPLACIANS IN NONSMOOTH DOMAINS},

year = {2012}

}

### OpenURL

### Abstract

In the first (and abstract) part of this survey 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 where S = −∆ | C ∞ 0 (Ω) in L2 (Ω; dnx) for Ω ⊂ Rn 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), SKv = λv, λ ̸ = 0, 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 formulas (−∆) 2 u = λ(−∆)u in Ω, λ ̸ = 0, u ∈ H 2 0 (Ω), u = S −1 F (−∆)v, v = λ−1 (−∆)u, with SF the Friedrichs extension of S. This establishes

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Citation Context ...tant function in L2 (R; dnx) and hence u ≡ 0 in Ω, establishing (10.8). □ Next, we record some useful capacity results. For an authoritative extensive discussion on this topic see the monographs [3], =-=[129]-=-, [163], and [180]. We denote by Bα,2(E) the Bessel capacity of order α > 0 of a set E ⊂ Rn . When K ⊂ Rn is a compact set, this is defined by Bα,2(K) := inf { ‖f‖ 2 L2 (Rn;dn ∣ x) gα ∗ f ≥ 1 on K, f ... |

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Citation Context ...onding spectral distribution function (i.e., the number of eigenvalues of −∆D,Ω not exceeding λ). The study of the asymptotic behavior of ND,Ω(λ) as λ → ∞ has been initiated by Weyl in 1911–1913 (cf. =-=[178]-=-, [177], and the references in [179]), in response to a question posed in 1908 by the physicist Lorentz, pertaining to the equipartition of energy in statistical mechanics. When n = 2 and Ω is a bound... |

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Citation Context ....42) the collection of eigenvalues for the perturbed Dirichlet Laplacian HD,Ω (again, listed according to their multiplicity). Then, if 0 ≤ V ∈ L∞ (Ω; dnx), we have the well-known formula (cf., e.g., =-=[63]-=- for the case where V ≡ 0) λD,Ω,j = min Wj subspace of H1 0 (Ω) ( dim(Wj)=j max 0̸=u∈Wj ) RD,Ω[u] , j ∈ N, (8.43) where RD,Ω[u], the Rayleigh quotient for the perturbed Dirichlet Laplacian, is given b... |

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52 | Sobolev spaces. Second edition - Adams, Fournier |

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Citation Context ...·) abbreviates the Hankel function of the first kind and order zero (cf., [1, Sect. 9.1]). Here the Donoghue-type Weyl–Titchmarsh operators (cf. [69] in the case where dim(N+) = 1 and [80], [82], and =-=[88]-=- in the general abstract case where dim(N+) ∈ N ∪ {∞}) MHF,Rn and MH \{0},N+ K,Rn are \{0},N+ defined according to equation (4.8) in [80]: More precisely, given a self-adjoint extension ˜ S of the den... |

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Citation Context ... on C ∞ 0 (Ω) as a nonnegative, self-adjoint operator in L 2 (Ω; d n x). The smallest (in the operator theoretic order sense) such realization has been introduced, in an abstract setting, by M. Krein =-=[116]-=-. Later it was realized that in the case where the symmetric operator, whose self-adjoint extensions are sought, has a strictly positive lower bound, Krein’s construction coincides with one that von N... |

44 |
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Citation Context ...elated not only to the geometrical properties of the domain (as well as the type of boundary conditions), but also to the smoothness properties of its boundary (the monograph by Safarov and Vassiliev =-=[156]-=- contains a wealth of information on this circle of ideas). These considerations are by no means limited to the Laplacian; see [57] for the case of the Stokes operator, and [38], [44] for the case the... |

42 |
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Citation Context ...al kernel of the resolvent (−∆ − z) −1 ), and H (1) 0 (·) abbreviates the Hankel function of the first kind and order zero (cf., [1, Sect. 9.1]). Here the Donoghue-type Weyl–Titchmarsh operators (cf. =-=[69]-=- in the case where dim(N+) = 1 and [80], [82], and [88] in the general abstract case where dim(N+) ∈ N ∪ {∞}) MHF,Rn and MH \{0},N+ K,Rn are \{0},N+ defined according to equation (4.8) in [80]: More p... |

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Citation Context ... refer to [83], [84], [85], [86], and [87]. For an extensive list of references on z-dependent Dirichlet-to-Neumann maps we also refer, for instance, to [7], [11], [15], [34], [47], [49], [50], [51], =-=[52]-=-, [64], [65], [81]–[87], [96], [147], [153], [154], [155]. 6. Regularized Neumann Traces and Perturbed Krein Laplacians This section is structured into two parts dealing, respectively, with the regula... |

26 |
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Citation Context ...the subject of self-adjoint extensions of semibounded operators (not necessarily restricted to the Krein–von Neumann extension) we refer to Birman [36], [37], Friedrichs [76], Freudenthal [75], Grubb =-=[91]-=-, [92], Krein [117], ˘ Straus [162], and Vi˘sik [171] (see also the monographs by Akhiezer and Glazman [10, Sect. 109], Faris [74, Part III], and the recent book by Grubb [97, Sect. 13.2]). An intrins... |

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Citation Context ...2 (R; dnx) and hence u ≡ 0 in Ω, establishing (10.8). □ Next, we record some useful capacity results. For an authoritative extensive discussion on this topic see the monographs [3], [137], [174], and =-=[191]-=-. We denote by Bα,2(E) the Bessel capacity of order α > 0 of a set E ⊂ Rn . When K ⊂ Rn is a compact set, this is defined by Bα,2(K) := inf { ‖f‖ 2 L2 (Rn;dn ∣ x) gα ∗ f ≥ 1 on K, f ≥ 0 } , (10.10) wh... |

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25 |
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Citation Context ...) follows. Formula (7.42), along with the final conclusion in the statement of the theorem, is also implicit in the above analysis plus the fact that ker(HK,Ω) is infinite-dimensional (cf. (2.46) and =-=[136]-=-). □ 8. Eigenvalue Estimates for the Perturbed Krein Laplacian The aim of this section is to study in greater detail the nature of the spectrum of the operator HK,Ω. We split the discussion into two s... |

24 | Weyl function and spectral properties of self-adjoint extensions, Integral Eqs. Operator Theory 43
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(Show Context)
Citation Context ...ith nonsmooth domains we refer to [83], [84], [85], [86], and [87]. For an extensive list of references on z-dependent Dirichlet-to-Neumann maps we also refer, for instance, to [7], [11], [15], [34], =-=[47]-=-, [49], [50], [51], [52], [64], [65], [81]–[87], [96], [147], [153], [154], [155]. 6. Regularized Neumann Traces and Perturbed Krein Laplacians This section is structured into two parts dealing, respe... |

24 |
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Citation Context ...is intriguing result (along with others, similar in spirit) has been obtained. Surprising connections between Weyl’s asymptotic formula and geometric measure theory have been explored in [56], [104], =-=[120]-=- for fractal domains. Collectively, this body of work shows that the nature of the Weyl asymptotic formula is intimately related not only to the geometrical properties of the domain (as well as the ty... |

24 |
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Citation Context ...) holds. Formulas (1.55)–(1.57) are also a byproduct of the connection between (1.60) and (1.61) and known spectral estimates for the buckling plate problem from [26], [27], [30], [59], [105], [141], =-=[143]-=-, [144]. Similarly, (1.58) for convex domains is based on the connection between (1.60) and (1.61) and the eigenvalue inequality relating the first eigenvalue of a fixed membrane and that of the buckl... |

24 | Lineare Operatoren in Hilberträumen, Teil 2 - Weidmann - 2003 |

23 |
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Citation Context ...icular, we wish to clarify the extent to which a Weyl asymptotic formula continues to hold for this operator. For us, this undertaking was originally inspired by the discussion by Alonso and Simon in =-=[14]-=-. At the end of that paper, the authors comment to the effect that “It seems to us that the Krein extension of −∆, i.e., −∆ with the boundary condition (1.35), is a natural object and therefore worthy... |

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23 |
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Citation Context ...here the symmetric operator, whose self-adjoint extensions are sought, has a strictly positive lower bound, Krein’s construction coincides with one that von Neumann had discussed in his seminal paper =-=[172]-=- in 1929. For the purpose of this introduction we now briefly recall the construction of the Krein–von Neumann extension of appropriate L2 (Ω; dnx)-realizations of the differential operator A of order... |

22 | Some applications of operator-valued Herglotz functions
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(Show Context)
Citation Context ...in and Dirichlet Laplacians Coincide Motivated by the special example where Ω = R2 \{0} and S = −∆C∞ 0 (R2\{0}), in which case one can show the interesting fact that SF = SK (cf. [12], [13, Ch. I.5], =-=[80]-=-, and Subsections 11.4 and 11.5) and hence the nonnegative self-adjoint extension of S is unique, the aim of this section is to present a class of (nonempty, proper) open sets Ω = Rn \K, K ⊂ Rn compac... |

22 |
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Citation Context ...8)–(8.60) have been proved in [25], [26], [27], [59], and [105] (indeed, further strengthenings of (8.59) are detailed in [26], [27]), whereas the respective parts of (8.61) are covered by results in =-=[114]-=- and [141] (see also [30], [46]). □ON THE KREIN–VON NEUMANN EXTENSION 43 Remark 8.10. Given the physical interpretation of the first eigenvalue for (8.57), it follows that λ (0) K,Ω,1 , the first non... |

21 | An Addendum to Krein’s formula
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Citation Context ... H (1) 0 (·) abbreviates the Hankel function of the first kind and order zero (cf., [1, Sect. 9.1]). Here the Donoghue-type Weyl–Titchmarsh operators (cf. [69] in the case where dim(N+) = 1 and [80], =-=[82]-=-, and [88] in the general abstract case where dim(N+) ∈ N ∪ {∞}) MHF,Rn and MH \{0},N+ K,Rn are \{0},N+ defined according to equation (4.8) in [80]: More precisely, given a self-adjoint extension ˜ S ... |

21 | On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains
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Citation Context ...omain with a compact boundary, we recall the existence of a universal linear extension operator EΩ : D ′(Ω) → S ′(Rn) such that EΩ : Hs (Ω) → Hs (Rn) is bounded for all s ∈ R, and RΩEΩ = IHs (Ω) (cf. =-=[152]-=-). If ˜ C∞ 0 (Ω) denotes the set of C∞ 0 (Ω)-functions extended to all of Rn by setting functions zero outside of Ω, then for all s ∈ R, ˜ C∞ 0 (Ω) ↩→ Hs 0(Ω) densely. Moreover, one has ( s H0(Ω) ) ∗ ... |

21 |
Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers
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(Show Context)
Citation Context ...) satisfies ˚H s (Ω) = H s z (Ω) if s > −1/2, s /∈ { } 1 2 + N0 . (4.12) For a Lipschitz domain Ω ⊆ Rn with compact boundary it is also known that ( ) s ∗ −s H (Ω) = H (Ω), −1/2 < s < 1/2. (4.13) See =-=[165]-=- for this and other related properties. Throughout this survey, we agree to use the adjoint (rather than the dual) space X ∗ of a Banach space X. From this point on we will always make the following a... |

19 | Isoperimetric and universal inequalities for eigenvalues, in ‘‘Spectral Theory and Geometry
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Citation Context ...λ (0) D,Ω,2 ≤ λ(0) K,Ω,1 . (8.61) Proof. With the eigenvalues of the buckling plate problem replacing the corresponding eigenvalues of the Krein Laplacian, estimates (8.58)–(8.60) have been proved in =-=[25]-=-, [26], [27], [59], and [105] (indeed, further strengthenings of (8.59) are detailed in [26], [27]), whereas the respective parts of (8.61) are covered by results in [114] and [141] (see also [30], [4... |

19 |
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Citation Context ...efer to [21]–[24]. For classical references on the subject of self-adjoint extensions of semibounded operators (not necessarily restricted to the Krein–von Neumann extension) we refer to Birman [36], =-=[37]-=-, Friedrichs [76], Freudenthal [75], Grubb [91], [92], Krein [117], ˘ Straus [162], and Vi˘sik [171] (see also the monographs by Akhiezer and Glazman [10, Sect. 109], Faris [74, Part III], and the rec... |

19 | Nonselfadjoint operators, infinite determinants, and some
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Citation Context ...4], [85], [86], and [87]. For an extensive list of references on z-dependent Dirichlet-to-Neumann maps we also refer, for instance, to [7], [11], [15], [34], [47], [49], [50], [51], [52], [64], [65], =-=[81]-=-–[87], [96], [147], [153], [154], [155]. 6. Regularized Neumann Traces and Perturbed Krein Laplacians This section is structured into two parts dealing, respectively, with the regularized Neumann trac... |

18 | M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems
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(Show Context)
Citation Context ...nsmooth domains we refer to [83], [84], [85], [86], and [87]. For an extensive list of references on z-dependent Dirichlet-to-Neumann maps we also refer, for instance, to [7], [11], [15], [34], [47], =-=[49]-=-, [50], [51], [52], [64], [65], [81]–[87], [96], [147], [153], [154], [155]. 6. Regularized Neumann Traces and Perturbed Krein Laplacians This section is structured into two parts dealing, respectivel... |

18 | Deficiency indices and singular boundary conditions in quantum mechanics - Bulla, Gesztesy - 1985 |

18 |
Variations on a theme of Jost and
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(Show Context)
Citation Context ... ∅. (4.59) − ∆D,Ω and − ∆N,Ω, (4.60) for HD,Ω and HN,Ω, respectively, and simply refer to these operators as, the Dirichlet and Neumann Laplacians. The above results have been proved in [81, App. A], =-=[87]-=- for considerably more general potentials than assumed in Hypothesis 4.7. Next, we shall now consider the minimal and maximal perturbed Laplacians. Concretely, given an open set Ω ⊂ R n and a potentia... |

17 |
Generalized Minkowski content, spectrum of fractal drums, fractal strings and Riemann zeta-function
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- 1997
(Show Context)
Citation Context ...here this intriguing result (along with others, similar in spirit) has been obtained. Surprising connections between Weyl’s asymptotic formula and geometric measure theory have been explored in [56], =-=[104]-=-, [120] for fractal domains. Collectively, this body of work shows that the nature of the Weyl asymptotic formula is intimately related not only to the geometrical properties of the domain (as well as... |

17 |
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(Show Context)
Citation Context ...he existence of convex domains Ω, for which the first eigenfunction of the problem of a clamped plate and the problem of the buckling of a clamped plate possesses a change of sign, was established in =-=[113]-=-. Relations between an eigenvalue problem governing the behavior of an elastic medium and the buckling problem were studied in [106]. Buckling eigenvalues as a function of the elasticity constant are ... |

16 | Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains
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- 2008
(Show Context)
Citation Context ..., z ∈ C\σ(HD,Ω), (5.46) MN,D,Ω,V (z) ∈ B ( N 1/2 (∂Ω) , N 3/2 (∂Ω) ) , z ∈ C\σ(HN,Ω). (5.47) For closely related recent work on Weyl–Titchmarsh operators associated with nonsmooth domains we refer to =-=[83]-=-, [84], [85], [86], and [87]. For an extensive list of references on z-dependent Dirichlet-to-Neumann maps we also refer, for instance, to [7], [11], [15], [34], [47], [49], [50], [51], [52], [64], [6... |

15 | Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions
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(Show Context)
Citation Context ...ators associated with nonsmooth domains we refer to [83], [84], [85], [86], and [87]. For an extensive list of references on z-dependent Dirichlet-to-Neumann maps we also refer, for instance, to [7], =-=[11]-=-, [15], [34], [47], [49], [50], [51], [52], [64], [65], [81]–[87], [96], [147], [153], [154], [155]. 6. Regularized Neumann Traces and Perturbed Krein Laplacians This section is structured into two pa... |

15 |
Positive selfadjoint extensions of positive symmetric operators
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(Show Context)
Citation Context ... hence, QS(f, g) = (f, SF g)H, f ∈ dom(QS), g ∈ dom(SF ), dom(QS) = dom ( (SF ) 1/2) . (2.19) An intrinsic description of the Krein–von Neumann extension SK of S ≥ 0 has been given by Ando and Nishio =-=[16]-=- in 1970, where SK has been characterized as the operator SK : dom(SK) ⊂ H → H given by SKu := S ∗ u, u ∈ dom(SK) := { v ∈ dom(S ∗ ) ∣ ∣ there exists {vj}j∈N ⊂ dom(S), (2.20) with lim j→∞ ‖Svj − S ∗ v... |

15 |
On the theory of self-adjoint extensions of positive definite operators
- Birman
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Citation Context ...s we refer to [21]–[24]. For classical references on the subject of self-adjoint extensions of semibounded operators (not necessarily restricted to the Krein–von Neumann extension) we refer to Birman =-=[36]-=-, [37], Friedrichs [76], Freudenthal [75], Grubb [91], [92], Krein [117], ˘ Straus [162], and Vi˘sik [171] (see also the monographs by Akhiezer and Glazman [10, Sect. 109], Faris [74, Part III], and t... |

15 | Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices
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Citation Context ...ins we refer to [83], [84], [85], [86], and [87]. For an extensive list of references on z-dependent Dirichlet-to-Neumann maps we also refer, for instance, to [7], [11], [15], [34], [47], [49], [50], =-=[51]-=-, [52], [64], [65], [81]–[87], [96], [147], [153], [154], [155]. 6. Regularized Neumann Traces and Perturbed Krein Laplacians This section is structured into two parts dealing, respectively, with the ... |