The use of point interactions makes the Schr\"{o}dinger equation of
quantum mechanics (say, on half-axis) exactly solvable (readers are
recommended to consult the comprehensive 2005 monograph [1] by
Albeverio et al). Once one assumes their special case of derivatives
of Dirac delta functions centered at a strictly increasing unbounded
sequence $X = \{x_k\}_{k=1}^\infty$ and endowed with respective
couplings $\beta_k$, one usually consults the Gesztesy's and
Holden's 1987 paper [18] for basic information. In this context, the
present authors propose to employ, in a way paralleling ref. [5],
the language of forms. Having resolved the key technical puzzle of
making the related forms closable (by splitting them in two at each
center $k$) they are able to spot the key differences between
$\delta$ and $\delta'$ point interactions emphasizing that in the
latter case [a] the role of a free ``unperturbed'' Hamiltonian is
taken by a direct sum of its Neumann's analogues in
$L^2(x_{k-1},x_k)$ (cf. also [16]), [b] one can parallel several
$\delta-$interaction results known from the Glazman's and
Wiedenholtz' monographs [19,40] and Molchanov's paper [36] deducing,
in particular, the self-adjointness from the semiboundedness. The
criteria of semiboundedness are then studied and used to find
several (though still not all) necessary and sufficient conditions
of the spectral discreteness and essentialness. As a premium,
multiple interesting cases of operators with exotic essential
spectra are finally discussed and sampled.

MR3165916 (Sent 2014-05-06) Kostenko, Aleksey; Malamud, Mark
Spectral theory of semibounded Schrödinger operators with
$\delta'$-interactions. Ann. Henri Poincaré 15 (2014), no. 3,
501--541. 47A10