The purpose of this chapter is to develop a framework for the description and study of hamiltonians having a many-channel structure. The term “many-channel” is used here in a rather vague sense: we are thinking of systems consisting of a (large, but finite) number of components which could interact in a complicated way but could also behave independently (i.e. the interaction between some components could be turned off). So, to the “total hamiltonian” H one should be able to associate a collection of “sub-hamiltonians” H a which, in some sense, should be simpler than H, and then one should construct the spectral and scattering theory of H in terms of the family {H a}.
The purpose of this chapter is to develop a framework for the description and study of hamiltonians having a many-channel structure. The term “many-channel” is used here in a rather vague sense: we are thinking of systems consisting of a (large, but finite) number of components which could interact in a complicated way but could also behave independently (i.e. the interaction between some components could be turned off). So, to the “total hamiltonian” H one should be able to associate a collection of “sub-hamiltonians” H a which, in some sense, should be simpler than H, and then one should construct the spectral and scattering theory of H in terms of the family {H a}.
So far, we have dealt with one- and two-body systems in a variety of settings. These are quite simple, at least a priori, since the equations needed to solve one- and twoparticle scattering problems, as well as bound states, are usually easy enough to handle. The situation becomes much more intricate in the three-body problem, as we shall see in this chapter. Firstly, while two particles in free space can either scatter or bind together, three bodies have extra channels (for attractive interactions) corresponding to two bound particles scattering off a third. Moreover, the energies of these extra channels can be degenerate with those of three unbound particles, and the total momentum of a pair within the triplet is not conserved. Fortunately, we will see that getting our hands dirty with the three-body problem pays off, since there are a number of exotic phenomena associated with the quantum three-body problem, for both bound and scattering states. In this chapter, we introduce the formalism necessary to investigate the three-body problem and study in detail some of the most striking phenomenology known to date associated with, at least, the simplest interactions.
We especially need imagination in science. aria Mitchell Summary The many-body problem consists of two parts: the first is the non-interacting system in a materials-specific external potential; the second is the Coulomb interaction that makes the problem so hard to solve. The most straightforward idea is to use perturbation theory, with the Coulomb interaction as perturbation. This is conceptually simple, but it turns out to be difficult in practice, since the Coulomb interaction is often not small compared with typical energy differences, it is long-ranged and in the thermodynamic limit there is an infinite number of particles, contributing with an infinite number of mutual interaction processes. The present chapter outlines how one can deal with this problem. It contains an overview of facts that one can also find in many standard textbooks on the many-body problem, but that are useful to keep in mind in order to look at later chapters from a sound and well-established perspective. The many-body problem is a tough one, and it has many facets. Sorting it out is like putting together a huge puzzle. The eight introductory chapters of this book provide pieces of the puzzle, and ideas on what one might do about it. In the present chapter we choose to go in one of the possible directions, in order to arrive at something tangible. The chapter gives the general framework and the main ideas; specific approximations are the topic of Chs. 10–15. The idea is to start from an independent-particle problem and add the Coulomb interaction as a perturbation. This is not easy: first, the interaction is responsible for a rich variety of phenomena that are completely absent otherwise, such as the finite lifetime of quasiparticles, or additional structures in spectra due to the fact that a quasi-particle excitation may transfer its energy to other elementary excitations, for example plasmons. Second, because of the two-body Coulomb interaction, the problem scales badly with the number of electrons, and straightforward perturbation theory for the many-body hamiltonian with the Coulomb interaction as perturbation rapidly becomes intractable or even useless, especially in large systems. To get started, Sec. 9.1 recalls why things are not so easy. The following sections try to solve one problem after the other, starting from Sec. 9.2 where the Green's function is reformulated in a way that is appropriate for a perturbation expansion.
Background There are two stages in the study of perturbation theory and related techniques (although they are mixed intimately in most derivations in the literature). The first is the formal development , carried out in terms of the total Hamiltonian and total wave function (and total zero-order wave function), without attempt to express anything in terms of one- and two-body quantities (components of Ĥ, orbitals, integrals over orbitals etc.). We can make a considerable amount of progress in this way before considering the detailed form of Ĥ. The second is the many-body development , where all expressions are obtained in terms of orbitals (one-electron states) and oneand two-electron integrals. We shall try to keep these separate for a while and begin with a consideration of formal perturbation theory. Another aspect of the study of many-body techniques is the large variety of approaches, notations and derivations that have been used. Each different approach has contributed to the lore and the language of many-body theory, and each tends to illuminate some aspects better than the other approaches. If we want to be able to read the literature in this field, we should be familiar with several alternative formulations. Therefore, we shall occasionally derive some results in more than one way and, in particular, we shall derive the basic perturbation-theory equations and their many-body representations in several complementary ways. Classical derivation of Rayleigh–Schrödinger perturbation theory The perturbation Ansatz We begin with a classical textbook derivation of formal Rayleigh–Schrödinger perturbation theory (RSPT).
We study spectral properties of quantum many-body Hamiltonians through a subsystem-based framework. Given a Hamiltonian of the form H=∑X⊆ΛΦ(X) acting on a tensor product Hilbert space, we associate to each subset S⊆Λ a subsystem Hamiltonian HS and its spectrum S(S)=σ(HS). This produces a family of spectra indexed by subsystems, allowing spectral data to be organized according to interaction structure. We show that subsystem Hamiltonians admit local approximations: HS can be approximated by operators supported on finite neighborhoods with an error bounded by ∥HS−HS,r∥≤∣S∣e−μr∥Φ∥μ. As a consequence, subsystem spectra are stable under truncation in the sense that dH(S(S),σ(HS,r))≤∣S∣e−μr∥Φ∥μ. We then prove that for disjoint subsets S1,S2⊆Λ, the subsystem spectrum is approximately additive: dH(S(S1∪S2),S(S1)+S(S2))≤(∣S1∣+∣S2∣)e−μD∥Φ∥μ, where D=d(S1,S2). In the finite-range case, this relation becomes exact. The results show that spectral properties reflect the locality of interactions not only at the level of operators, but also at the level of spectra. The framework provides a way to study many-body systems in which interaction geometry directly shapes spectral behavior.
A non-Hermitian matrix Hamiltonian H appears in the wavefunction form of a variety of many-body scattering theories. This operator acts on an arrangement channel Banach or Hilbert space 𝒞 = ⊕αℋ where ℋ is the N-particle Hilbert space and α are certain arrangement channels. Various aspects of the spectral and semigroup theory for H are considered. The normalizable and weak (wavelike) eigenvectors of H are naturally characterized as either physical or spurious. Typically H is scalar spectral and ’’equivalent’’ to H on an H-invariant subspace of physical solutions. If the eigenvectors form a basis, by constructing a suitable biorthogonal system, we show that H is scalar spectral on 𝒞. Other concepts including the channel space observables, trace class and trace, density matrix and Möller operators are developed. The sense in which the theory provides a ’’representation’’ of N-particle quantum mechanics and its equivalence to the usual Hilbert space theory is clarified.
In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrödinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate r1,r2,... and momentum p1, p2, . . . , boson creation and annihilation operators, b†iα, biα. The index i runs over the number of relevant degrees of freedom, while the index α runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i ≠ j, . . . [biα, b†jα´] = 0, [biα, bjα´] = 0,. . . . . .[bjα, b†iα´] = 0, [b†jα, b†iα´] = 0,. . . . . . [biα, b†iα´] = ẟαα´, [biα, b†iα´] = 0, [b†iα, b†iα´] = 0. . . .