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Journal of Physics A Mathematical and General

On the quantum mechanics of complex Hamiltonian systems in one dimension

R. S. Kaushal

Published Dec 5, 2001• 37 citations
Publisher

Abstract

The solutions of an analogous Schrödinger wave equation for the one-dimensional non-Hermitian Hamiltonian H ( x , p ) in the complex phase plane characterized by x = x 1 + i p 2 , p = p 1 + i x 2 , are investigated. The quasi-exact solutions thus obtained reveal a lot about the nature of the complex eigenvalue spectrum of the potential concerned. The examples of harmonic, harmonic plus inverse harmonic and Mörse potentials are discussed.

DOI
10.1088/0305-4470/34/49/104

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The solutions of an analogous Schrödinger wave equation for the one-dimensional non-Hermitian Hamiltonian H ( x , p ) in the complex phase plane characterized by x = x 1 + i p 2 , p = p 1 + i x 2 , are investigated. The quasi-exact solutions thus obtained reveal a lot about the nature of the complex eigenvalue spectrum of the potential concerned. The examples of harmonic, harmonic plus inverse harmonic and Mörse potentials are discussed.
Within the framework of an extended complex phase space approach characterized by x = x 1 + i p 2 , p = p 1 + i x 2 , we investigate the quasi-exact solutions of an analogous Schrödinger equation for a variety of complex sextic potentials of the type V ( x ) = a + bx + cx 2 + dx 3 + ex 4 + fx 5 + hx 6 with complex couplings. It is found that the eigenvalue spectrum in general is complex for complex couplings and real for the real parameters of the potential. While the ground state solutions for an even power sextic potential with an inverse harmonic term and for the - symmetric version of V ( x ) are obtained, other variants of V ( x ) are also investigated with reference to the excited states.
With a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one-dimensional Schrödinger equation corresponding to a non-Hermitian Hamiltonian H ( x , p ) we investigate the ground-state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x = x 1 + i p 2 , p = p 1 + i x 2 , where ( x 1 , p 1 ) and ( x 2 , p 2 ) are real and considered as canonical pairs. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems. It is noted that the imaginary part of the eigenvalue, E i , turns out to be zero for all potentials V ( x ) with real couplings whereas it turns out to be nonzero for the case when the couplings are complex. The prescription is also extended to study the excited states. The problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed.
With a view to getting further insight into the solutions of one-dimensional analogous Schr?dinger equation for a non-hermitian (complex) Hamiltonian system, we investigate the quasi-exact PT- symmetric solutions for an octic potential and its variant using extended complex phase space approach characterized by x=x1+ip2, p=p1+ix2, where (x1, p1) and (x2, p2) are real and considered as canonical pairs. Besides the complexity of the phase space, complexity of potential parameters is also considered. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalue and eigenfunction of a system. The imaginary part of energy eigenvalue of a non-hermitian Hamiltonian exist for complex potential parameters and reduces to zero for real parameters. However, in the present work, it is found that imaginary component of the energy eigenvalue vanishes even when potential parameters are complex, provided that PT-symmetric condition is satisfied. Thus PT- symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue.
We investigate the quasi-exact solutions of the Schr?dinger wave equation for two-dimensional non-hermitian complex Hamiltonian systems within the frame work of an extended complex phase space characterized by x = x1 + ip3, y = x2 + ip4, px = p1 + ix3, py = p2 + ix4. Explicit expressions of the energy eigenvalues and the eigenfunctions for ground and first excited states for a complex quartic potential are obtained. Eigenvalue spectra of some variants of the complex quartic potential, including PT-symmetric one, are also worked out.
For gaining further insight into the nature of the eigenspectra of a complex octic potential [say], we investigate the quasi exact solutions of the Schr?dinger equation in an extended complex phase space characterized by . The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunction of a system. Explicit expressions of eigenvalues and eigenfunctions for the ground state as well as for the first excited state of a complex octic potential and its variant are worked out. It is found that imaginary part of the eigenvalue turns out to be zero for real coupling parameters, whereas it becomes non-zero for complex coupling parameters. However, the PT-symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue even if coupling parameters in the potential are complex.
We present exact solutions of the Schrödinger equation for a quantum system with complex mass subjected to a complex Morse potential in the extended complex phase space. The normalized eigenfunctions and corresponding eigenspectra are derived within a non-Hermitian framework, ensuring consistent probability densities. Conditions for the reality of the spectra are established and used to analyze the dependence of eigenvalue behaviour on potential parameters. The study reveals distinct regimes of spectral characteristics arising from the interplay of complex mass, the Morse parameter, and eigenvalues, leading to the emergence of five classes of non-Hermitian quantum states. By analysing the energy eigenspectra, normalization conditions, and probability density profiles across parameter space, we identify regimes corresponding to real-spectrum Hermitian-like matter, quasi-stable or resonant states, purely complex quantum matter, non- physical, non-normalizable states, and a quasi-classical determinate regime in which the probability density becomes spatially static. One of these systems exhibits a non-dissipative, collisionless state with long-range gravitational-like characteristics, suggesting a theoretical analogue for dark matter within a non-Hermitian quantum framework. Further, the five identified classes of matter may be interpreted as distinct matter phases of a single quantum system governed by complex mass and Morse parameters. This classification elucidates the boundary between physical and non-physical regimes in complex quantum systems and provides a unified approach for interpreting stability, resonance, and emergent classicality arising from complex parameters.