Periodic Potential from the Spectral Invariants

Abstract

In this chapter, we consider the inverse problem of the three-dimensional Schrödinger operator $L(q)$ with a periodic, relative to a lattice $\Omega$ of $\mathbb {R}^{3},$ potential $q.$ Firstly, we construct a set $D$ of trigonometric polynomials such that: (a) $D$ is dense in $W_{2}^{s}(\mathbb {R}^{3}/\Omega ),$ where $s>3,$ in the $\mathbb {C}^{\infty }$-topology, (b) any element $q$ of the set $D$ can be determined constructively and uniquely, modulo inversion $x\rightarrow -x$ and translations $x\rightarrow x+\tau$ for $\tau \in \mathbb {R}^{3}$, from the given spectral invariants that were determined constructively from the given Bloch eigenvalues. Then a special class $V$ of the periodic potentials is constructed, which can be easily and constructively determined from the spectral invariants . This chapter consists of 7 sections. First section is introduction, where we describe briefly the scheme of this chapter and discuss the related papers. In the second section using the spectral invariants obtained in Chap. 3 we find the simplest invariants for the sets $D$ and $V$. In the third, fourth and fifth sections we give algorithms for the unique determination of the potential $q\in D$ and $q\in V$ respectively from the simplest spectral invariants. In the sixth section we consider the stability of the algorithm for $q\in V$ with respect to the spectral invariants and Bloch eigenvalues. Finally, in the seventh section we prove that there are no other periodic potentials in the set of large class of functions whose Bloch eigenvalues coincide with the Bloch eigenvalues of $q\in V.$ Thus, Chap. 4 gives some examples and ideas for finding the potential from the spectral invariants and hence from the Bloch eigenvalues. Besides it gives a theoretical base (a lot of nonlinear equations with respect to the Fourier coefficients of $q$) to solve numerically this problem.