In this chapter we begin to study the solutions of the electronic Schrodinger equation and compile and prove some basic, for the most part well-known, facts about its solutions in suitable form. Parts of this chapter are strongly influenced by Agmon’s monograph [3] on the exponential decay of the solutions of second-order elliptic equations. Starting point are two constants associated with the solution spaces introduced in the previous chapter, the minimum energy that the given system can attain and the ionization threshold. Both constants are intimately connected with the spectral properties of the Hamilton operator and are introduced in the first section of this chapter. The second section deals with some notions and simple results from spectral theory that are rewritten here in terms of bilinear forms as they underly the weak form of the Schrodinger equation. The weak form of the equation will not only be the starting point of the regularity theory that we will develop later, but is also the basis for many approximation methods of variational type, from the basic Ritz method discussed in the third section to the many variants and extensions of the Hartree-Fock method. Our exposition is based on simple, elementary properties of Hilbert spaces like the projection theorem, the Riesz representation theorem, or the fact that every bounded sequence contains a weakly convergent subsequence. Hence only a minimum of prerequisites from functional analysis is required. For a comprehensive treatment of spectral theory and its application to quantum mechanics, we refer to texts like [44, 69–71], or [87, 88]. We finally show, in the fourth section, that the essential spectrum of the electronic Schrodinger operator is non-empty and that the ionization threshold represents its lower bound. We will assume that the minimum energy is located below the ionization threshold. It is then an eigenvalue, the ground state energy. The corresponding eigenfunctions are the ground states. The knowledge of the ground states and particularly of the ground state energy is of main interest in quantum chemistry. The last section is devoted to the exponential decay of the eigenfunctions for eigenvalues below the ionization threshold, a result that goes back to O’Connor [20] and has later been substantially refined [3]. In contrast to many other presentations the symmetry properties of the wave functions enforced by the Pauli principle are hereby carefully taken into account.