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In the absence of a third dredge-up episode during the asymptotic giantbranch phase, white dwarf models evolved from low-metallicity progenitors havea thick hydrogen envelope, which makes hydrogen shell burning be the mostimportant energy source. We investigate the pulsational stability of whitedwarf models with thick envelopes to see whether nonradial $g$-mode pulsationsare triggered by hydrogen burning, with the aim of placing constraints onhydrogen shell burning in cool white dwarfs and on a third dredge-up during theasymptotic giant branch evolution of their progenitor stars. We constructwhite-dwarf sequences from low-metallicity progenitors by means of fullevolutionary calculations, and analyze their pulsation stability for the modelsin the range of effective temperatures $T_{\rm eff} \sim 15\,000\,-\, 8\,000$K. We demonstrate that, for white dwarf models with masses $M_{\star} \lesssim0.71\,\rm M_{\sun}$ and effective temperatures $8\,500 \lesssim T_{\rm eff}\lesssim 11\,600$ K that evolved from low-metallicity progenitors ($Z= 0.0001$,$0.0005$, and $0.001$) the dipole ($\ell= 1$) and quadrupole ($\ell=2$) $g_1$modes are excited mostly due to the hydrogen-burning shell through the$\varepsilon$-mechanism, in addition to other $g$ modes driven by either the$\kappa-\gamma$ or the convective driving mechanism. However, the $\varepsilon$mechanism is insufficient to drive these modes in white dwarfs evolved fromsolar-metallicity progenitors. We suggest that efforts should be made toobserve the dipole $g_1$ mode in white dwarfs associated with low-metallicityenvironments, such as globular clusters and/or the galactic halo, to placeconstraints on hydrogen shell burning in cool white dwarfs and the thirddredge-up episode during the preceding asymptotic giant branch phase.

Pulsations powered by hydrogen shell burning in white dwarfs