Effective quantum Hamiltonian in thin domains with non-homogeneity

This work aims to derive an effective model of the Laplacian with a non-homogeneous metric in thin domains with Neumann boundary conditions. Firstly, the Neumann Laplace operator with a non-homogeneous failure will be defined as a self-adjoint operator on the Hilbert space by an associated quadratic form. Furthermore, this work shows the convergence of this operator to the effective model in the spectral, the strong resolvent, even in the norm-resolvent sense, all of which are illustrated with a concrete example. Finally, the rate of the convergence is derived.