## 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

### MSC:

 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