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Register a new userThe Krein-von Neumann extension revisited
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We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.
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We discuss the Krein--von Neumann extensions of three Laplacian-type operators -- on discrete graphs, quantum graphs, and domains. In passing we present a class of one-dimensional elliptic operators such that for any $nin mathbb N$ infinitely many elements of the class have $n$-dimensional null space.
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, $Sgeq epsilon I_{mathcal{H}}$ for some $epsilon >0$ in a Hilbert space $mathcal{H}$ to an abstract buckling problem operator. In the concrete case where $S=bar{-Delta|_{C_0^infty(Omega)}}$ in $L^2(Omega; d^n x)$ for $Omegasubsetmathbb{R}^n$ 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 $S_K$ (i.e., the Krein--von Neumann extension of $S$), [ S_K v = lambda v, quad lambda eq 0, ] is in one-to-one correspondence with the problem of {em the buckling of a clamped plate}, [ (-Delta)^2u=lambda (-Delta) u text{in} Omega, quad lambda eq 0, quad uin H_0^2(Omega), ] where $u$ and $v$ are related via the pair of formulas [ u = S_F^{-1} (-Delta) v, quad v = lambda^{-1}(-Delta) u, ] with $S_F$ 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.).
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, $Sgeq varepsilon I_{mathcal{H}}$ for some $varepsilon >0$ in a Hilbert space $mathcal{H}$ to an abstract buckling problem operator. 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.). In the second, and principal part of this survey, we study spectral properties for $H_{K,Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^infty_0(Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $Omegasubsetmathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.
This article constructs a surface whose Neumann-Poincare (NP) integral operator has infinitely many eigenvalues embedded in its essential spectrum. The surface is a sphere perturbed by smoothly attaching a conical singularity, which imparts essential spectrum. Rotational symmetry allows a decomposition of the operator into Fourier components. Eigenvalues of infinitely many Fourier components are constructed so that they lie within the essential spectrum of other Fourier components and thus within the essential spectrum of the full NP operator. The proof requires the perturbation to be sufficiently small, with controlled curvature, and the conical singularity to be sufficiently flat.
For an arbitrary open, nonempty, bounded set $Omega subset mathbb{R}^n$, $n in mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we consider the closed, strictly positive, higher-order differential operator $A_{Omega, 2m} (a,b,q)$ in $L^2(Omega)$ defined on $W_0^{2m,2}(Omega)$, associated with the higher-order differential expression $$ tau_{2m} (a,b,q) := bigg(sum_{j,k=1}^{n} (-i partial_j - b_j) a_{j,k} (-i partial_k - b_k)+qbigg)^m, quad m in mathbb{N}, $$ and its Krein--von Neumann extension $A_{K, Omega, 2m} (a,b,q)$ in $L^2(Omega)$. Denoting by $N(lambda; A_{K, Omega, 2m} (a,b,q))$, $lambda > 0$, the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K, Omega, 2m} (a,b,q)$, we derive the bound $$ N(lambda; A_{K, Omega, 2m} (a,b,q)) leq C v_n (2pi)^{-n} bigg(1+frac{2m}{2m+n}bigg)^{n/(2m)} lambda^{n/(2m)} , quad lambda > 0, $$ where $C = C(a,b,q,Omega)>0$ (with $C(I_n,0,0,Omega) = |Omega|$) is connected to the eigenfunction expansion of the self-adjoint operator $widetilde A_{2m} (a,b,q)$ in $L^2(mathbb{R}^n)$ defined on $W^{2m,2}(mathbb{R}^n)$, corresponding to $tau_{2m} (a,b,q)$. Here $v_n := pi^{n/2}/Gamma((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in $mathbb{R}^n$. 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 $widetilde A_{2} (a,b,q)$ in $L^2(mathbb{R}^n)$. We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension $A_{F,Omega, 2m} (a,b,q)$ in $L^2(Omega)$ of $A_{Omega, 2m} (a,b,q)$. No assumptions on the boundary $partial Omega$ of $Omega$ are made.
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