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.
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.
The time-independent scattering theory associated with the non-self-adjoint matrix Hamiltonians H of arrangement channel quantum mechanics is presented in detail first using the 3-particle case as an example. A key feature is the biorthogonality of a suitably constructed set of scattering eigenvectors and duals. Channel space Möller operators, S- and T-matrices are defined and a variety of properties investigated including the way multichannel unitarity is embedded into the theory. Some remarks on the time-dependent theory are also made. A detailed discussion of channel space density matrix scattering theory (of interest, e.g., in reactive kinetic theory) is presented using the Liouville representation. We describe some special cases including the exclusion of breakup and 2×2 choices of three particle H.
Quantum mechanics can be formulated either in terms of matrix mechanics or in terms of wave mechanics. This chapter centers on a discussion of the historical origins, mathematical structure, and extension to non-Hermitian quantum theory of matrix mechanics. The development of atomic discrete spectra and the old quantum theory showed that classical mechanics could not account for the discreteness of atomic energy levels and spectral lines, which motivated Heisenberg, Born, Jordan, and others to establish matrix mechanics centered on noncommuting physical quantities. Combining the Bohr atomic model, angular momentum quantization, and radiative transitions, one can see how matrix mechanics describes the energy level structure and quantum transitions of atomic systems, and how the Heisenberg equation of motion gives the fundamental form of quantum dynamics. The example of the harmonic oscillator further shows that the discreteness of the energy spectrum can be obtained directly from the matrix method, thereby establishing a clear connection among transition frequencies, matrix elements, and radiation-induced transitions. The matrix representation corresponds to the operator theory in Hilbert space, where state vectors, inner products, complete bases, and spectral decomposition constitute the fundamental mathematical framework of quantum mechanics. For closed quantum systems, physical observables are generally required to be represented by Hermitian or self-adjoint operators, so as to ensure that their eigenvalues possess the physically acceptable property of being real numbers. The Hermiticity of an operator is also closely related to its domain and boundary conditions, which provides a mathematical possibility for the emergence of non-Hermitian Hamiltonians. Based on these discussions, this chapter further explains how non-Hermitian dynamics is connected with open systems, dissipative processes, and special boundary conditions, thereby demonstrating the natural connection between traditional matrix mechanics and modern non-Hermitian quantum theory.
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.
Derivations are given for some transition and resolvent operator equations for multichannel quantum scattering with short-range potentials. The basic difference between these and previous equations is that the unknown operators act only on the channel subspaces. This is made possible by utilizing, and extending, the two-Hilbert-space formulation previously given by the authors [in J. Math. Phys. 14, 1328 (1973)]. The equations in abstract form are of the Lippmann–Schwinger type, differing only in the appearance of certain injection operators from one Hilbert space to the other. When applied to multichannel quantum scattering, the abstract theory yields a new system of equations for the transition and resolvent operators. Uniqueness of the solution to the equations is proved.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.
After a fairly rigorous mathematical formulation of quantum mechanics based on transformation theory by Dirac in which he cast Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics on a common operator theoretic setting and successfully explained their equivalence, it was John Von Neumann in monumental treatise “Mathematical foundations of quantum mechanics” who proved the spectral theorem for all self adjoint operators in an infinite dimensional Hilbert space and thereby cast the masterly work of Dirac “The principles of quantum mechanics“ in a way that even pure mathematicians were able to understand. Dirac in his book had also indicated the work of Jordan and Heisenberg on quantum scattering theory but without the modern method of wave operators. Lippmann and Schwinger gave a more rigorous discussion of scattering theory via the introduction of free particle states and scattered states. Actually, free particle states are not square integrable and hence cannot strictly be termed as states. However by the use of wave operators the equations of Lippmann and Schwinger could be made rigorous using the setting of the continuous spectrum of a self-adjoint opearator introduced by Von Neumann, Crucial to this development is the notion of the domain of definition of an unbounded operator. This leads to the fact that the wave operators that appear in scattering theory are not defined on all vectors but usually on those vectors that belong to the continuous spectrum of the free particle Hamiltonian. Today scattering theory is a well developed subject and the interested reader can get a good grasp of this subject from the books 1 Tosio Kato, “Perturbation theory for linear operators”, Springer. 2 Werner O. Amrein, “Hilbert space methods in quantum mechanics”, CRC press, 2012.