On the evaluation of the group of eigenfunctions and spectral function of the Laplace operator multiplied by a piecewise constant coefficient and the Dirichlet problem in the three-dimensional domain

The article examines the problem on the eigenfunctions of the Laplace operator multiplied by a piecewise constant coefficient and the Dirichlet problem in the three-dimensional domain. The evaluation of the group of eigenfunctions in a closed region (the so-called "bundle" in the terminology of V.A. Ilyin, i.e. a part of the spectral function when both points coincide) and the asymptotics of the spectral function when one point is on the surface of the coefficient discontinuity and the other is outside this surface are obtained. The estimate and asymptotics are found to the accuracy of the logarithmic multiplier. From the proved estimation and asymptotics, a uniform estimation of the group of eigenfunctions in the whole closed region and a uniform asymptotic estimation of the spectral function are obtained.