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No abstract uploaded!

This text is a part of an unfinished project which deals with the generalized point interaction (GPI) in one dimension. We employ two natural parametrizations, which are known but have not attracted much attention, to express the resolvent of the GPI Hamiltonian as well as its spectral and scattering properties. It is also shown that the GPI yields one of the simplest models in which a non-trivial Berry phase is exhibited. Furthermore, the generalized Kronig-Penney model corresponding to the GPI is discussed. We show that there are three different types of the high-energy behaviour for the corresponding band spectrum.

No abstract uploaded!

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).