Galerkin formulation of 1D linear Schrodinger equation

I am interested in the weak (Galerkin) formulation of the 1D Schrodinger equation:

$i u_t−\beta u_{xx}=0$ and $u(x,0)=u_0(x)$. As usual, we do integration by parts which yields:

$i (u_t,v)+\beta (u_x,v_x)=0$ and the boundary terms vanish by appropriate testing with test functions $v$. We then have the continuous sesquilinear form $b(u,v)=i (u_t,v)+\beta (u_x,v_x)$. I would like to apply the generalized Lax-Milgram on $b(\cdot,\cdot)$, but I am having trouble showing the boundedness below (coercivity) of $b(\cdot,\cdot)$. The coefficient $\beta$ is assumed to be real.

Is there a slick way to show $b(u,u)\geq \alpha \|u\|_{H^1}^2$? What I am stuck on is the complex coefficient which destroys the positivity of the form $b(\cdot,\cdot)$.

Many thanks in advance.

• I like your question. It has been a while since I did any quantum mechanics, but could you not avoid the problem all together by studying the time independent Schrödinger equation instead? – Carl Christian Feb 10 '16 at 11:58
• I need stability in space-time, so I can't do time independent – Sriram Nagaraj Feb 10 '16 at 18:14