%  a

\documentstyle[12pt]{article}
%\setlength{\parindent}{5mm}
% \renewcommand{\baselinestretch}{1.5}
\setlength{\headheight}{0pt}
 \setlength{\headsep}{0pt}
\setlength{\footskip}{45pt}
 \setlength{\footheight}{0pt}
 \setlength{\textwidth}{430pt}
 %d%
 % \setlength{\textheight}{250pt}
 %%  \setlength{\textheight}{350pt}
%%%%%%  \setlength{\textheight}{450pt}
 %vp%
 \setlength{\textheight}{600pt}
 \setlength{\oddsidemargin}{10pt}
 \def\ha{\mbox{$\frac{1}{2}$}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\ba{\begin{array}{c}}
\def\ea{\end{array}}
\def\p{\partial}
\def\ben{\[}
\def\een{\]}
\begin{document}

Spectral theory enjoys finding connections between  classical vibrations and
quantum Schroedigner operators. A particularly popular class of illustrations
involves the relation between oscillations of strings and various
delta-function-type solvable descendants of the quantum Kronig-Penney model
(cf. Kronig R de L and Penney W G 1931 Proc. R. Soc. A 130 499). A sample of
these results is provided in the four-page paper in question, the subject of
which is inspired by the incompleteness of equivalence between the spectra of
the Krein's inhomogeneous strings (with positive masses -- cf. refs. [5] - [7])
and derivative-delta-function quantum oscillators (without positivity
constraint, i.e., mathematically more general and methodically more
challenging). After a ``warm-up" theorem 1 (showing that n
derivative-delta-functions produce n energy levels), theorems 2 and 3 (on the
same interaction but with respective support on the Cantor discontinuum and on
an equidistant point set in the infintely-many-point limit) is more
mind-boggling since the only limiting point of the ``bound state energies"
proves to be minus infinity.


MR1988013

Nizhnik, L. P. A Schrödinger operator with $\delta'$-interaction. (Russian)
Funktsional. Anal. i Prilozhen. 37 (2003), no. 1, 85--88.




\end{document}