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A heterostructure composed of two parallel homogeneous layers is studied inthe limit as their widths $l_1$ and $l_2$, and the distance between them $r$shrinks to zero simultaneously. The problem is investigated in one dimensionand the squeezing potential in the Schr\"{o}dinger equation is given by thestrengths $V_1$ and $V_2$ depending on the layer thickness. A whole class offunctions $V_1(l_1)$ and $V_2(l_2)$ is specified by certain limitcharacteristics as $l_1$ and $l_2$ tend to zero. The squeezing limit of thescattering data $a(k)$ and $b(k)$ derived for the finite system is shown toexist only if some conditions on the system parameters $V_j$, $l_j$, $j=1,2$,and $r$ take place. These conditions appear as a result of an appropriatecancellation of divergences. Two ways of this cancellation are carried out andthe corresponding two resonance sets in the system parameter space are derived.On one of these sets, the existence of non-trivial bound states is proven inthe squeezing limit, including the particular example of the squeezed potentialin the form of the derivative of Dirac's delta function, contrary to thewidespread opinion on the non-existence of bound states in $\delta'$-likesystems. The scenario how a single bound state survives in the squeezed systemfrom a finite number of bound states in the finite system is described indetail.

Scattering data and bound states of a squeezed double-layer structure