Multidimensional distribution functions for elliptic operators (Q1337914)

Let \[ A_j'= {1\over{(2\pi)^k}} \int_{\mathbb{R}^k} \Biggl(\int_{\mathbb{R}^k} e^{is(x-y)} a_j' \biggl({{x+y}\over 2},s\biggr) u(y)dy \Biggr) dx, \qquad j=1,\dots,n \] be Weyl pseudodifferential operators with nonnegative symbols \(a_j'(x,s)\) which satisfy some conditions. Let \(\Omega\subset \mathbb{R}^k\) be an open set of finite measure and \(\chi_\Omega\) the operator of multiplication by the characteristic function of \(\Omega\). Consider the operators \[ \widetilde{A}_j= \chi_\Omega A_j'u, \qquad D(\widetilde{A}_j)= C_0^\infty(\Omega), \qquad j=1,\dots,n, \] in \(L_2(\Omega)\). Let \(B_j=b_j(x,D_x)\) \((j=1,\dots,n)\) be differential operators with coefficients of class \(C^\infty (\overline{\Omega})\) and \(A_j\geq0\) \((j=1,\dots,n)\) be some selfadjoint extensions of \(\widetilde {A}_j\) in \(L_2(\Omega)\). The author considers the asymptotics of the multidimensional distribution function \[ F(\lambda)=\text{sp}\,B_1 E_1(\lambda_1)\dots B_nE_n(\lambda_n), \] where \(\lambda=(\lambda_1,\dots, \lambda_n)\in \mathbb{R}^n\), \(E_j(\mu)\) is the spectral projector of \(A_j\).