J. F. Brasche

Let $\mathcal E$ and $\mathcal P$ be nonnegative quadratic forms such that $\mathcal E + b \mathcal P$ is closed and densely defined for every nonnegative real number $b$. Let $H_b$ be the selfadjoint operator associated with $\mathcal E + b\mathcal P.$ By Kato's monotone convergence theorem, there exists an operator $L$ such that $(H_b+1)^{-1}$ converges to $L$ strongly, as $b$ tends to infinity. We give a condition which is sufficient in order that the operators $(H_b+1)^{-1}$ converge w.r.t. the trace norm with convergence rate $O(1/b)$. As an application we discuss trace norm resolvent convergence of Schrodinger operators with point interactions.

We consider various forms of boundary-value conditions for general one-dimensional Schrödinger operators with point interactions that include $\delta$-- and $\delta'$-- interactions, $\delta'$-- potential, and $\delta$-- magnetic potential. We give most simple spectral properties of such operators, and consider a possibility of finding their norm resolvent approximations.

We consider examples of generalized selfadjoint operators that act from a positive Hilbert space to a negative space. Such operators were introduced and studied in [1]. We give examples of selfadjoint operators on the principal Hilbert space $H_ 0$ that, being considered as operators from the positive space $H_ + \subset H_ 0$ into the negative space $H_ - \supset H_ 0$, are not essentially selfadjoint in the generalized sense.