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Let$L$be a formally selfadjoint differential operator and$p$a real-valued function, both on$a$≤$x$<∞. The deficiency indices are the numbers of solutions of$Lu$=λ$pu$for Im λ>0 and for Im λ<0 which have a certain regularity at$x$=∞. (A) If$p(x)$≥0 this regularity means that the integral of$p(x)$│$u$│^{2}converges at infinity. (B) If$p$changes its sign for arbitrarily large values of$x$but$L$has a positive definite Dirichlet integral it is natural to relate the regularity to this integral. Weyl’s classical study of the deficiency indices is reviewed for (A) with the help of elementary theory of quadratic forms. Individual bounds are found for the deficiency indices also when$L$is of odd order. It is then indicated how the method carriers over to (B).

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The aim of the paper is to derive spectral estimates into several classes of magnetic systems. They include three-dimensional regions with Dirichlet boundary as well as a particle in 3 confined by a local change of the magnetic field. We establish two-dimensional BerezinLiYau and LiebThirring-type bounds in the presence of magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians.