  We investigate the spectrum of a soft quantum waveguide in two dimensions of
the generalized `bookcover' shape, that is, Schr\"odinger operator with the
potential in the form of a ditch consisting of a finite curved part and
straight asymptotes which are parallel or almost parallel pointing in the same
direction. We show how the eigenvalues accumulate when the angle between the
asymptotes tends to zero. In case of parallel asymptotes the existence of a
discrete spectrum depends on the ditch profile. We prove that it is absent in
the weak-coupling case, on the other hand, it exists provided the transverse
potential is strong enough. We also present a numerical example in which the
critical strength can be assessed.
