  Based on group theoretical arguments we derive the most general Hamiltonian
for the $\text{Bi}_2\text{Se}_3$-class of materials including terms to third
order in the wave vector, first order in electric and magnetic fields, first
order in strain and first order in both strain and wave vector. We determine
analytically the effects of strain on the electronic structure of
$\text{Bi}_2\text{Se}_3$. For the most experimentally relevant surface
termination we analytically derive the surface state spectrum, revealing an
anisotropic Dirac cone with elliptical constant energy counturs giving rise to
different velocities in different in-plane directions. The spin-momentum
locking of strained $\text{Bi}_2\text{Se}_3$ is shown to be modified and for
some strain configurations we see a non-zero spin component perpendicular to
the surface. Hence, strain control can be used to manipulate the spin degree of
freedom via the spin-orbit coupling. We show that for a thin film of
$\text{Bi}_2\text{Se}_3$ the surface state band gap induced by coupling between
the opposite surfaces changes opposite to the bulk band gap under strain.
Tuning the surface state band gap by strain, gives new possibilities for the
experimental investigation of the thickness dependent gap and optimization of
optical properties relevant for, e.g., photodetector and energy harvesting
applications. We finally derive analytical expressions for the effective mass
tensor of the Bi$_2$Se$_3$ class of materials as a function of strain and
electric field.
