Spectral theory of semibounded Schr\"odinger operators with ${\delta }^{\mathrm{\prime }}$-interactions

We study spectral properties of Hamiltonians ${\text{rH}}_{X,\text{gB},q}$ with ${\delta }^{\mathrm{\prime }}$-point interactions on a discrete set $X={x_k}_{k=1}^{\mathrm{\infty }}\subset {\mathbb{R}}_{+}$. %at the centers $x_n$ on the positive half line in terms of energy forms. Using the form approach, we establish analogs of some classical results on operators ${\text{rH}}_q=-d^2\mathrm{/}d{x}^2+q$ with locally integrable potentials $q\in L_{\text{loc}}^1({\mathbb{R}}_{+})$. In particular, we establish analogues of the Glazman-Povzner-Wienholtz theorem, the Molchanov discreteness criterion, and the Birman theorem on stability of an essential spectrum. It turns out that in contrast to the case of Hamiltonians with $\delta$-interactions, spectral properties of operators ${\text{rH}}_{X,\text{gB},q}$ are closely connected with those of ${\text{rH}}_{X,q}^N={\oplus }_k{\text{rH}}_{q,k}^N$, where ${\text{rH}}_{q,k}^N$ is the Neumann realization of $-d^2\mathrm{/}d{x}^2+q$ in $L^2(x_{k-1},x_k)$.