*Book Chapter*

**Inserted:** 2 jun 2017

**Last Updated:** 21 apr 2018

**Year:** 2017

**Links:**
Link to the chapter

**Abstract:**

This is a chapter of the book "Shape optimization and spectral theory". We consider Schrödinger operator of the form $-\Delta+V(x)$ on the Sobolev space $H^1_0(D)$, where $D$ is a bounded open subset of ${\mathbb R}^d$. We are interested in finding optimal potentials for some suitable criteria; the optimization problems we deal with are then written as $$\min\big\{F(V)\ :\ V\in{\mathcal V}\big\}$$ where $F$ is a suitable cost functional, typical example being $F(V)=\Phi\big(\lambda_{1}(V),\dots,\lambda_{k}(V)\big),$ where $\Phi$ is a given function and $\lambda_{k}(V)$ are the eigenvalues of the operator $-\Delta+V$.