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We investigate scaling phenomena at first-order quantum transitions, when theboundary conditions favor one of the two phases. We show that the correspondingfinite-size scaling behavior, arising from the interplay between the drivingparameter and the finite size of the system, is more complex than that emergingwhen boundary conditions do not favor any phase. We discuss this issue in theframework of the paradigmatic one-dimensional quantum Ising model, along itsfirst-order quantum transition line driven by an external longitudinal field.

Finite-size scaling at first-order quantum transitions when boundary conditions favor one of the two phases