Spectra of Hamiltonians

Abstract

The chapter is involved with several aspects concerning the spectra of Hamiltonians. In §4.1, §4.3, §4.4, the nature of the spectra of some general classes of Hamiltonian is determined under sufficiency conditions satisfied by the underlying potentials. These sufficiency conditions are readily verified and if satisfied, the nature of the spectra may be inferred. This is obviously very important if the exact solution of the problem in question is not known. Even if the solution may be explicitly obtained, the knowledge of the nature of the spectrum of a Hamiltonian prior to its determination is quite useful. Bound-states are studied in §4.2. In this section sufficiency conditions are given for their existence as well as for their absence for various physical systems. In §4.5, we carry out an analysis for counting the eigenvalues (if any) of given Hamiltonians. This investigation is then used in §4.6 to derive lower bounds to the expectation values of kinetic energy operators for singleand many-particle systems. The results obtained in these two sections will find important applications to multi-electron atoms in Chapter 13 and to the problem of the stability of matter in Chapter 14. The final section (§4.7) deals with the role of supersymmetry in solving the eigenvalue problem and in the construction of supersymmetric Hamiltonians. In this section, we make use of the general properties of supersymmetry transformations obtained in §2.9.