Isospectral mapping for quantum systems with energy point spectra to polynomial quantum harmonic oscillators

Abstract: We show that a polynomial H(N) of degree N of a harmonic oscillator hamiltonian allows us to devise a fully solvable continuous quantum system for which the first N discrete energy eigenvalues can be chosen at will. In general such a choice leads to a re-ordering of the associated energy eigenfunctions of H such that the number of their nodes does not increase monotonically with increasing level number. Systems H have certain universal features, we study their basic behaviours.

• The paper introduces a method to design quantum systems with predetermined energy eigenvalues using a polynomial of the harmonic oscillator Hamiltonian.
• It demonstrates that employing a polynomial mapping can reorder energy spectra, challenging traditional nodal patterns dictated by the Sturm-Liouville theorem.
• The study presents computational techniques that ensure stability in the numerical construction of Hamiltonians while enabling scalability for moderate energy spectra.

Isospectral Mapping for Quantum Systems

This paper explores the concept of isospectral mapping in quantum mechanical systems, particularly focusing on the transformation of energy spectra from general bound-state systems to that of polynomial quantum harmonic oscillators. The primary objective is to construct quantum systems whose energy levels are predetermined, using a polynomial Hamiltonian of the quantum harmonic oscillator. This study primarily contributes to the theoretical understanding of quantum mechanics, providing insights into the energy spectrum design and manipulation of wave functions.

Polynomial Hamiltonian Construction

The paper introduces a method for creating a continuous one-dimensional quantum system with predefined energy levels using a polynomial of the harmonic oscillator Hamiltonian. By employing real polynomials $\hat{\cal H}_{(N)}$ of degree $N$, the authors demonstrate that one can design a Hamiltonian where the first $N$ energy eigenvalues can be selected arbitrarily. This construction challenges conventional spectral ordering, as the associated energy eigenstates do not necessarily follow the Sturm-Liouville theorem's node count rule.

Mathematical Framework

The construction relies on the polynomial form:

$$\hat {\cal H}_{(N)} \equiv {\cal P}_{(N)} (\hat h) = \sum_{j=1}^N a_j \hat h^j$$

where $\hat h$ is the Hamiltonian of a dimensionless harmonic oscillator. The polynomial mapping allows the spectrum of $\hat{h}$, which is linear, to be transformed into a desired spectrum while having all wave functions derived from the harmonic oscillator, hence maintaining full solvability.

Spectral Reordering and Violations

A key result of the paper is that mapping through $\hat {\cal H}_{(N)}$ can lead to reordering of energy eigenfunctions, thereby allowing the construction of Hamiltonians where the node count does not increase monotonically with energy level. This is a notable deviation from what is normally expected in quantum spectral theory, primarily concerning second-order differential equations.

Dialing Spectra at Will

The methodology allows practitioners to specify the point spectrum explicitly, generating Hamiltonians whose energy eigenstates fit any prescribed sequence of real numbers. The paper illuminates that higher order terms in the polynomial result in kinetic energy contributions, which lead to coefficients that do not converge as $N \to \infty$, but instead oscillate and grow, thus preventing straightforward convergence of the Hamiltonian.

Practical Implications and Computational Techniques

The study provides computational techniques using exact fractions to avoid instabilities that typically arise due to numerical approximations. This ensures stability in designing these isospectral transformations, even when dealing with polynomials of high degree. The scope is extended to explore multi-well potential systems and the inherent biases in choosing the order $N$ to retain desirable spectral properties.

Stability and Scalability

The construction demonstrates that as the potential number of energy levels mapped increases, the polynomial order required also increases, bringing into question practical scalability. Nonetheless, the framework is applicable for moderate numbers of mapped energy levels, making it relevant for physical systems modeled by polynomial Hamiltonians.

Phase Space Dynamics

The paper investigates the phase space characteristics, specifically the behavior of the Wigner distribution and its current, emphasizing that while some systems do allow phase space current alignment with energy contours, complex quantum mechanical systems generally do not align as such due to inherent singularities.

Conclusions

In conclusion, this paper presents a substantial theoretical advancement in quantum spectrum manipulation, offering a robust mathematical toolset for crafted energy levels in quantum systems. It provides a method to construct a broad class of Hamiltonians with customizable spectral properties, offering a new angle on the universal treatment of discrete spectrum quantum systems. Future work may expand on interacting multi-particle systems or further explore the implications of spectral reordering for quantum state engineering.