Quasi-isospectral higher-order Hamiltonians via a reversed Lax pair construction
arXiv:2507.19622 · doi:10.1088/1402-4896/ae5853
Abstract
We present a novel approach for constructing quasi-isospectral higher-order Hamiltonians from time-independent Lax pairs by reversing the conventional interpretation of the Lax pair operators. Instead of treating the typically second-order -operator as the Hamiltonian, we take the higher-order -operator as the starting point and construct a sequence of quasi-isospectral operators via intertwining techniques. This procedure yields a variety of new higher-order Hamiltonians that are isospectral to each other, except for at least one state. We illustrate the approach with explicit examples derived from the KdV equation and its extensions, discussing the properties of the resulting operators based on rational, hyperbolic, and elliptic function solutions. In some cases, we present infinite sequences of quasi-isospectral Hamiltonians, which we generalise to shape-invariant differential operators capable of generating such sequences. Our framework provides a systematic mechanism for generating new integrable systems from known Lax pairs.
16 pages
References in corpus (12)
- Self-isospectrality, special supersymmetry, and their effect on the band structure
- Hidden supersymmetry in quantum bosonic systems
- Spectral singularities in PT-symmetric periodic finite-gap systems
- Classical and Quantum Dynamics of Higher-Derivative Systems
- Finite-gap systems, tri-supersymmetry and self-isospectrality
- Existence of Different Intermediate Hamiltonians in Type A N-fold Supersymmetry
- Higher time-derivative theories from space-time interchanged integrable field theories
- Higher derivative Hamiltonians with benign ghosts from affine Toda lattices
- Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials
- Integrable scattering theory with higher derivative Hamiltonians
- Unusual isospectral factorizations of shape invariant Hamiltonians with Scarf II potential
- Nonlinear evolution of disturbances in higher time-derivative theories