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Two low-mass pre-white dwarfs, which could be precursors of ELM white dwarfs,have been observed to show multiperiodic photometric variations. They couldconstitute a new class of pulsating low-mass pre-white dwarf stars. We presenta detailed nonadiabatic pulsation study of such stars, employing fullevolutionary sequences of low-mass He-core pre-white dwarf models. We haveconsidered models in which element diffusion is accounted for and also modelsin which it is neglected. We confirm and explore in detail a new instabilitystrip in the domain of low gravities and low effective temperatures of the$T_{\rm eff}-\log g$ diagram, where low-mass pre-white dwarfs are currentlyfound. The destabilized modes are radial and nonradial $p$ and $g$ modesexcited by the $\kappa-\gamma$ mechanism acting mainly at the zone of thesecond partial ionization of He, with non-negligible contributions from theregion of the first partial ionization of He and the partial ionization of H.The computations with element diffusion are unable to explain the pulsationsobserved in the two known pulsating pre-white dwarfs, suggesting that elementdiffusion might be inhibited at these stages of the pre-white dwarf evolution.Our nonadiabatic models without diffusion, on the other hand, naturally explainthe existence and range of periods of the pulsating pre-white dwarf star WASPJ1628$+$10B, although they fail to explain the pulsations of WASP J0247$-$25B,the other known member of the class, indicating that the He abundance in thedriving region of this star might be substantially larger than predicted by ourmodels. Further discoveries of additional members of this new class ofpulsating stars and their analysis in the context of the theoretical backgroundpresented in this paper will shed new light on the evolutionary history oftheir progenitor stars.

Pulsating low-mass white dwarfs in the frame of new evolutionary sequences: III. The pre-ELM white dwarf instability strip