For which regular expressions $\alpha$ is $\{ \beta \mid L(\alpha) = L(\beta) \}$ PSPACE-complete?
This question is addressed in Section 2 of [1], which shows (Theorem 2.6) that the problem is


in P if $L(\alpha)$ is finite;
coNP-complete if $L(\alpha)$ is infinite but bounded (i.e. $L(\alpha)\subseteq w_1^*w_2^*\ldots w_k^*$ for some $w_1,\ldots, w_k$);
PSPACE-complete otherwise.


[1] Harry B. Hunt, Daniel J. Rosenkrantz, Thomas G. Szymanski, On the equivalence, containment, and covering problems for the regular and context-free languages, Journal of Computer and System Sciences, Volume 12, Issue 2, 1976, Pages 222-268, ISSN 0022-0000, http://dx.doi.org/10.1016/S0022-0000(76)80038-4.
(http://www.sciencedirect.com/science/article/pii/S0022000076800384)