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Register a new userSome remarks on the Krein--von Neumann extension of different Laplacians
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Delio Mugnolo
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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.
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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.
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 $C^{1,r}$, $r>1/2$.
We study spectral properties for $H_{K,Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-Delta+V$ 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 $C^{1,r}$, $r>1/2$. In particular, in the aforementioned context we establish the Weyl asymptotic formula [ #{jinmathbb{N} | lambda_{K,Omega,j}leqlambda} = (2pi)^{-n} v_n |Omega| lambda^{n/2}+Obig(lambda^{(n-(1/2))/2}big) {as} lambdatoinfty, ] where $v_n=pi^{n/2}/ Gamma((n/2)+1)$ denotes the volume of the unit ball in $mathbb{R}^n$, and $lambda_{K,Omega,j}$, $jinmathbb{N}$, are the non-zero eigenvalues of $H_{K,Omega}$, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein--von Neumann extension of $-Delta+V$ defined on $C^infty_0(Omega)$) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980s. Our work builds on that of Grubb in the early 1980s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting.
In his paper from 1996 on quadratic forms Heath-Brown developed a version of circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight. The weight function is assumed to be $C_0^infty$-smooth and to vanish near the singularity of the quadric. In out work we allow the weight function to be finitely smooth and not vanish near the singularity, and we give also an explicit dependence on the weight function.
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