Topics in spectral theory of differential operators
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This dissertation is devoted to two eigenvalue counting problems: Determining the asymptotic behavior of large eigenvalues of self-adjoint extensions of partial differential operators, and computing the number of negative eigenvalues for bounded from below operators with compact resolvents. In the first part of this thesis we derive a Weyl-type asymptotic formula and a bound for the eigenvalue counting function for the Krein-von Neumann extension of differential operators on open bounded subsets of R n. In the second part of this thesis we obtain a formula relating the Maslov index, a topological invariant counting the signed number of conjugate points of paths of Lagrangian planes in H1/2 (??) x H-1/2 (??), and the Morse index, the number of negative eigenvalues, for the second order differential operators with domains of definition contained in H1 (?) for open bounded subsets ? ? R n.