We consider boundary value problems for the one-dimensional 
Schr\"odinger operator with Dirac delta potential. 
Green functions $G(x,y)$ are constructed by using the 
symmetric orthogonalization method, and 
their aspects as reproducing kernel are also investigated. 
As an application, the best constants of the 
corresponding Sobolev inequalities is expressed as the  
maximum of the diagonal value $G(y,y)$.