Spectral theory for Schr\"odinger operators with $\delta$-interactions   supported on curves in $\mathbb R^3$
  The main objective of this paper is to systematically develop a spectral and scattering theory for selfadjoint Schr\"odinger operators with $\delta$-interactions supported on closed curves in $\mathbb R^3$. We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive Schatten--von Neumann properties for the resolvent difference with the free Laplacian, and establish an explicit representation for the scattering matrix. 
Jussi Behrndt, Rupert L. Frank, Christian K\"uhn, Vladimir   Lotoreichik, and Jonathan Rohleder
math.SP math-ph math.MP