I'm trying to understand how exactly $L_\sigma$ are defined. I noticed that if I take $\sigma=0$ then I get $B_0(u,v)=\langle Lu,v \rangle$, with respect to the standard $L^2$ inner product (and $L$ is the original Schrodinger operator). This seems like a possible direction on understanding how the operators are defined, but since they mentioned the space $H^1_0$ (which uses a different inner product), this might not be a correct idea.
I also found out that one can define self-adjoint operators from symmetric bilinear forms using the Riesz representation theorem, as described here. But this seems like a not very concrete way to define the $L_\sigma$ operators. If this is indeed the case, then I'm wondering if there's a more concrete way to interpret these operators (can we find a more concrete formula or something?), and once again, I'm not sure exactly if the writers are referring to the $L^2$ inner product or the $H^1_0$ inner product. Moreover, what's the relation to the original Schrodinger operator $L$, and why is the limit $L_\infty$ a Dirichlet boundary condition operator?
Anyway, if anyone has any ideas and can explain to me how to think of the $L_\sigma$ operators, I'd appreciate it. The original article is Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map, by Berkolaiko, Cox and Marzuola.
Thanks in advance!