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arXiv:1603.02005 · Journal of Physics A Mathematical and Theoretical

Non-self-adjoint Hamiltonians with complex eigenvalues

F Bagarello

Published Apr 26, 2016• 8 citations
arXivHTMLPDF

Abstract

Abstract Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.

DOI
10.1088/1751-8113/49/21/215304

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Abstract Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.
This paper is devoted to the construction of what we will call exactly solvable models, i.e., of quantum mechanical systems described by an Hamiltonian H whose eigenvalues and eigenvectors can be explicitly constructed out of some minimal ingredients. In particular, motivated by PT-quantum mechanics, we will not insist on any self-adjointness feature of the Hamiltonians considered in our construction. We also introduce the so-called bicoherent states, we analyze some of their properties and we show how they can be used for quantizing a system. Some examples, both in finite and in infinite-dimensional Hilbert spaces, are discussed.
In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H *. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H , is real but it is not a necessary condition. In the literature on so-called PT -symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs.
Update: A time-independent n×nn\times nn×n PT-symmetric (and symmetric) Hamiltonian is diagonalizable since it has all distinct real eigenvalues and the resulting diagonal matrix is a real symmetric matrix. The diagonalization results an isometry so there shouldn't be any issue with unitarity and unfortunately this very elementary mathematical fact somehow did not draw the authors' attention. However, PT-symmetric quantum mechanics is not out of trouble. For time-dependent PT-symmetric (and symmetric) Hamiltonians (even 2×22\times 22×2 ones) the authors observed that there is a violation of unitarity. Moreover, the first named author showed in his recent article arXiv:1312.7738 that PT-symmetric quantum mechanics is indeed a certain kind of Hermitian quantum mechanics and that in order for time-evolution to be unitary with respect to JJJ-inner product (one that gives rise to a Hilbert space structure on the space of state functions), the potential energy operator V(x)V(x)V(x) must be real. This means that those complex PT-symmetric Hamiltonians that have been studied by physicists are unfortunately unphysical. The first named author discussed in a subsequent article arXiv:1401.5149 that while finite-state PT-symmetric quantum mechanics with time-independent Hamiltonians is not physically any different from Hermitian quantum mechanics, PT-symmetric quantum mechanics exhibits a distinctive symmetry from that of Hermitian quantum mechanics.
Non-self-adjoint quantum mechanical operators do not necessarily possess eigenvalues. Finite N ? N matrix representations of these operators, however, can be hermitian and therefore have a finite set of N real eigenvalues. Using the momentum operator, the kinetic energy operator, and the relativistic Hamiltonian of the Coulomb problem for the Klein?Gordon equation as examples, we examine analytically and also numerically the properties of the spectrum and eigenvectors in finite dimensional Hilbert spaces. We study the limit of N ? ? for which some eigenvalues cease to exist as the corresponding operators are not self-adjoint.
In this article, we review the quantum mechanical setting associated with a non-self-adjoint Hamiltonian with a real spectrum. The spectral properties of the Hamiltonian of a Swanson-like model are investigated. The eigenfunctions associated with the real simple eigenvalues are shown to form complete systems but not a (Riesz) basis, which gives rise to difficulties in the rigorous mathematical formulation of quantum mechanics. A new inner product, appropriate for the physical interpretation of the model, has been consistently introduced. The dynamics of the system is described. Some specificities of the theory of non-self-adjoint operators with implications in quantum mechanics are discussed.
If a quantum-mechanical Hamiltonian is PT symmetric, there are two possibilities: either all of the eigenvalues are real, in which case the Hamiltonian is said to be in an unbroken-PT-symmetric phase, or else some of the eigenvalues are real and some are complex, in which case the Hamiltonian is said to be in a broken-PT-symmetric phase. As one varies the parameters of the Hamiltonian, one can pass through the phase transition that separates the unbroken and broken phases. This transition has recently been observed in a variety of laboratory experiments. This paper explains the phase transition in a simple and intuitive fashion and then describes an elementary experiment in which the phase transition is easily observed.