Spectral asymptotics for the ”soft” selfadjoint extension of a symmetric elliptic differential operator. (English) Zbl 0559.47035

Let A be a strongly elliptic formally selfadjoint partial differential operator of order \(k=2m\) \((m>0)\), defined on a bounded smooth domain \(\Omega\) in \(R^ n\) by \[ Au=\sum_{| \alpha | \leq 2m}a_{\alpha}(x)D^{\alpha}u,\quad a_{\alpha}(x)\in C^{\infty}({\bar \Omega}), \] where \(D^{\alpha}=(-i\partial /\partial x_ 1)^{\alpha_ 1}...(-i\partial /\partial x_ n)^{\alpha_ n}\) if \(\alpha =(\alpha_ 1,...,\alpha_ n)\). Let \(A_{\min}\) and \(A_ M\) respectively denote the minimal operator associated with A and the so- called soft (or Krein or von Neumann) extension of \(A_{\min}\). For any \(t>0\), let \(N(t;A_ M)\) be the number of eigenvalues of \(A_ M\) in the interval (0,t).
In this paper the author proves the asymptotic formula \[ N(t;A_ M)- c_ At^{n/2m}=O(t^{(n-\theta)/2m})\quad as\quad t\to \infty, \] where \(c_ A\) is the same constant as for the Dirichlet problem and \(\theta =\max \{1/2-\epsilon,2m/(2m+n-1)\}\) for any \(\epsilon >0\). This gives an affirmative answer to a question posed by A. Alonso and B. Simon [ibid. 4, 251-270 (1980; Zbl 0467.47017)].
Reviewer: A.Torgašev


47F05 General theory of partial differential operators
47A20 Dilations, extensions, compressions of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
35J30 Higher-order elliptic equations
47E05 General theory of ordinary differential operators


Zbl 0467.47017