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Physica Scripta

A quantum system with a non-self-adjoint 2D-harmonic oscillator

Natália Bebiano, João da Providência, João P. da Providência

Published Apr 24, 2019• 2 citations
Publisher

Abstract

Abstract Non-self-adjoint operators with a real discrete spectrum exhibit, in general, a quite different behavior from the one of self-adjoint operators, with the same spectrum. This fact has serious mathematical and physical implications, namely in the rigorous formulation of quantum mechanics (QM), where self-adjointness plays a central role, in particular in the spectral analysis of the involved operators. In this framework, a non-self-adjoint 2D-harmonic oscillator is considered. Its eigenvalues and eigenfunctions, as well as those of its adjoint, are explicitly determined. Although the eigensystems are complete they may not form bases, a drawback for the mathematical formulation of the QM of the system. The existence of a metric operator defining an inner product which renders the Hamiltonian self-adjoint, is discussed.

DOI
10.1088/1402-4896/ab1bf0

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Abstract Non-self-adjoint operators with a real discrete spectrum exhibit, in general, a quite different behavior from the one of self-adjoint operators, with the same spectrum. This fact has serious mathematical and physical implications, namely in the rigorous formulation of quantum mechanics (QM), where self-adjointness plays a central role, in particular in the spectral analysis of the involved operators. In this framework, a non-self-adjoint 2D-harmonic oscillator is considered. Its eigenvalues and eigenfunctions, as well as those of its adjoint, are explicitly determined. Although the eigensystems are complete they may not form bases, a drawback for the mathematical formulation of the QM of the system. The existence of a metric operator defining an inner product which renders the Hamiltonian self-adjoint, is discussed.
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.
Using as a prototype example the harmonic oscillator we show how losing self-adjointness of the Hamiltonian HHH changes drastically the related functional structure. In particular, we show that even a small deviation from strict self-adjointness of HHH produces two deep consequences, not well understood in the literature: First of all, the original orthonormal basis of HHH splits into two families of biorthogonal vectors. These two families are complete but, contrarily to what often claimed for similar systems, none of them is a basis for the Hilbert space H\mathcal{H}H. Second, the so-called metric operator is unbounded, as well as its inverse. In the second part of the paper, after an extension of some previous results on the so-called D\mathcal{D}D pseudobosons, we discuss some aspects of our extended harmonic oscillator from this different point of view.
Consider in L2(Rd)L^2(R^d)L2(Rd), d≥1d\geq 1d≥1, the operator family H(g):=H0+igWH(g):=H_0+igWH(g):=H0​+igW. $\ds H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2$ is the quantum harmonic oscillator with rational frequencies, WWW a PPP symmetric bounded potential, and ggg a real coupling constant. We show that if ∣g∣<ρ|g|<\rho∣g∣<ρ, ρ\rhoρ being an explicitly determined constant, the spectrum of H(g)H(g)H(g) is real and discrete. Moreover we show that the operator $\ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1$ has real discrete spectrum but is not diagonalizable.
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.
Quantum mechanics of a linear harmonic oscillator in a finite-dimensional Hilbert space satisfying the correct equations of motion is studied. The connections to Weyl's formulation of the algebra of bounded unitary operators in finite space as well as to a truncated quantized linear harmonic oscillator are discussed. It is pointed out that the discrete Fourier transformation (DFT) plays a central role in determining the actual form of the position, the momentum, the number and the Hamiltonian operators. The explicit form of these operators in different bases is exhibited for some low values of the dimension of the Hilbert space. In this formulation, it is shown that the Hamiltonian is indeed the logarithm of the DFT and that by modifying Weyl's framework to include position and momentum operators with non-uniformly spaced spectra the equations of motion are satisfied.
A basic fact, having fundamental significance in quantum mechanics, is that hermitian (or self-adjoint) operators have only real eigenvalues. However, in certain applications in molecular physics, one deals with non-hermitian operators. We discuss a condition for non-hermitian operators to have real eigenvalues, proving that it is the case if and only if it can be decomposed as a product of two, generally non-commuting hermitian operators, one of which is positive definite. The theorem is illustrated on the example of non-hermitian effective Hamiltonians occurring in the non-perturbative form of the Bloch equation.