Asymmetric Factorization Method Applied to Supersymmetry with Complex Operators: A Numerical Study

Core Concepts

This paper presents a novel method for constructing supersymmetric Hamiltonians using complex, PT-symmetric operators, demonstrating its validity through numerical calculations of energy levels for specific examples.

Abstract

This research paper investigates supersymmetry (SUSY) in quantum mechanics, specifically focusing on extending the concept to complex operators.

• Traditionally, SUSY is explored with real operators, where partner Hamiltonians (H+, H-) are generated from a superpotential (W) and their energy levels are related.
• This paper proposes an asymmetric factorization method for SUSY involving complex operators, ensuring they satisfy the necessary PT-symmetry condition for real energy spectra.
• The authors define annihilation and creation operators (A, B) using two different complex, PT-symmetric superpotentials (W1, W2), leading to the construction of partner Hamiltonians.
• The ground state wave function and relationships between energy levels of the partner Hamiltonians are derived.
• The paper provides specific examples of W1 and W2, illustrating the method and calculating the energy levels numerically using a matrix diagonalization technique.
• The obtained real energy spectra for the example complex potentials validate the proposed asymmetric factorization method.

The paper concludes that the proposed method successfully extends SUSY to complex potentials, confirmed by numerical results. This opens up possibilities for further exploration of SUSY in systems with complex operators, potentially impacting areas like PT-symmetric quantum mechanics.

Stats

The energy levels of H(-) are 0, 3.2867406, 10.4033155, 19.9364750.
The energy levels of H(+) are 3.2867406, 10.4033155, 19.9364750.

Quotes

"Supersymmetry is a powerful technique for realizing real spectra, which has been highly studied under real operators."
"However, the same has not been formulated using complex operators."
"In this paper, we develop supersymmetry for complex potentials using asymmetric formulation."

Key Insights Distilled From

Asymmetric factorization method on supersymmetry: Complex operators

by Biswanath Ra... at arxiv.org 11-06-2024

https://arxiv.org/pdf/1505.05018.pdf

Deeper Inquiries

How does this asymmetric factorization method compare to other techniques for handling complex potentials in quantum mechanics, beyond the scope of SUSY?

The asymmetric factorization method presented leverages the principles of SUSY, specifically the intertwining relations between partner Hamiltonians, to analyze complex potentials. However, other techniques exist within quantum mechanics to handle such potentials, each with strengths and limitations. Let's compare the asymmetric factorization method with some prominent alternatives:

Complex Scaling Method: This widely used method rotates the spatial coordinate or momentum into the complex plane, effectively transforming the Hamiltonian. While powerful for analyzing resonances and scattering states in complex potentials, it might not directly offer the elegance of SUSY in revealing spectral relationships between partner Hamiltonians.

Numerical Diagonalization: Techniques like the matrix diagonalization method mentioned in the paper are versatile and can handle a broad class of potentials. However, they might not provide the analytical insights and elegance offered by factorization methods, especially concerning the intertwining of energy levels.

Perturbation Theory: For complex potentials that can be expressed as small deviations from solvable real potentials, perturbation theory can offer approximate solutions. However, its accuracy depends on the strength of the perturbation and might not be suitable for strongly coupled complex potentials.

Other Factorization Methods: Beyond SUSY, other factorization techniques like the Darboux transformation exist. These methods might offer alternative approaches to construct solvable complex potentials and reveal spectral properties, potentially complementing or contrasting with the asymmetric factorization method.
In summary, the asymmetric factorization method, rooted in SUSY, provides a specific framework for analyzing complex potentials by exploiting the intertwining relations between partner Hamiltonians. While other techniques offer alternative approaches, the choice of method depends on the specific problem, desired insights, and computational resources.

Could there be scenarios where the proposed method fails to produce a real energy spectrum, and if so, what conditions might lead to such breakdowns?

While the asymmetric factorization method, as presented, aims to generate real energy spectra for complex potentials satisfying specific conditions, certain scenarios could lead to breakdowns, resulting in a non-real spectrum. Some potential causes include:

PT-Symmetry Breaking: The method relies on the PT-symmetry of the Hamiltonian. If the chosen complex potentials, specifically the superpotentials W1 and W2, do not satisfy the PT-symmetry condition ([PT, W1] = [PT, W2] = 0), the resulting Hamiltonian might not have a purely real spectrum.

Non-analyticity of Superpotentials: The derivation assumes a certain level of smoothness and analyticity for the superpotentials W1 and W2. If these functions exhibit singularities, branch cuts, or other non-analytic behavior within the relevant domain, the factorization method and the intertwining relations might break down, leading to a complex spectrum.

Spectral Singularities: Even when the Hamiltonian is PT-symmetric, specific energy levels might become "exceptional points" where the eigenstates coalesce and the Hamiltonian becomes non-diagonalizable. These points can lead to a breakdown of the usual SUSY intertwining relations and result in a complex energy spectrum.

Boundary Conditions: The choice of boundary conditions can significantly influence the spectrum. If the imposed boundary conditions are incompatible with the PT-symmetry of the potential or lead to a non-Hermitian problem, the energy spectrum might not be purely real.
In essence, the asymmetric factorization method's success in generating a real energy spectrum relies on a delicate interplay of PT-symmetry, analyticity of the superpotentials, and appropriate boundary conditions. Deviations from these conditions could lead to a breakdown of the method and result in a complex energy spectrum.

If we consider the complex operators as representing some physical observables, what implications might the observed energy level relationships have on our understanding of those physical systems?

Interpreting complex operators as representing physical observables is a subtle point, often encountered in PT-symmetric quantum mechanics. While directly associating them with measurable quantities might not always be straightforward, the observed energy level relationships can still offer valuable insights into the physical system's behavior.

Non-Hermitian but Real Spectrum: The fact that a non-Hermitian Hamiltonian (due to complex operators) can still possess a real energy spectrum suggests the presence of a hidden symmetry, often linked to PT-symmetry. This symmetry can impose constraints on the system's dynamics and lead to observable consequences, even if the operators themselves are not directly measurable.

Energy Level Intertwining: The observed intertwining of energy levels between the partner Hamiltonians (H+ and H-) implies a connection between their respective physical states. This connection could manifest as a specific relationship between transition probabilities, decay rates, or other dynamical properties of the system.

Analogies to Open Systems: Complex potentials are often used to model open quantum systems, where the imaginary part represents gain or loss. The real energy spectrum obtained through this method might suggest the existence of stable states or quasi-stable modes even in the presence of such non-Hermitian interactions.

Phase Transitions: The breakdown of PT-symmetry, often accompanied by the emergence of complex energies, can be interpreted as a phase transition in the system. This transition might correspond to a change in the system's stability, a shift in its dynamical behavior, or the emergence of new physical phenomena.
In conclusion, while directly interpreting complex operators as physical observables requires careful consideration, the observed energy level relationships, particularly the real spectrum and intertwining, provide valuable clues about the underlying physics. These relationships can point towards hidden symmetries, connections between different physical states, and potential phase transitions in the system, enriching our understanding of non-Hermitian and open quantum systems.