Inverse Spectral Theory and Differential Operators

Inverse spectral theory addresses the fundamental question of how to reconstruct a differential operator from the knowledge of its spectral characteristics. Central to this theory are operators such as the Sturm–Liouville, Schrödinger and Dirac operators, as well as their higher‐order generalisations. The spectrum of such an operator—its collection of eigenvalues and associated norming constants or spectral measures—encodes information about coefficients often referred to as potentials. By analysing this spectral data, one can recover the potential function or other defining parameters, a process of profound importance in mathematical physics, geometry and engineering. Applications range from determining quantum mechanical potentials to inferring material properties in vibration analysis and reconstructing the connectivity of networks modelled by metric graphs. Over recent decades the field has moved beyond classical self‐adjoint problems to treat non‐self‐adjoint operators, operators with distributional coefficients, and operators featuring nonlocal or delayed arguments, thereby widening its reach to complex systems and singular structures.

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Inverse Spectral Theory and Differential Operators publication trend

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Technical terms

Inverse spectral problem: The task of recovering an operator or its coefficients from knowledge of its spectrum and ancillary spectral data.

Sturm–Liouville operator: A second‐order differential operator defined by a potential function and boundary conditions, fundamental in spectral theory.

Spectrum: The set of eigenvalues of an operator, often associated with resonant frequencies or energy levels.

Distribution coefficient: A generalised function appearing in the definition of a differential operator, allowing for singular or measure‐valued potentials.

Frozen argument: A feature of certain operators in which coefficients depend on the function value at a shifted or delayed argument.

Weyl matrix: A matrix‐valued function encoding boundary behaviour and spectral characteristics of higher‐order operators.

References

1. Spectral analysis of the indefinite non-self-adjoint Sturm–Liouville operator. Partial Differential Equations in Applied Mathematics (2024).
2. Reconstruction of Differential Operators with Frozen Argument. Axioms (2022).
3. Reconstruction of Higher-Order Differential Operators by Their Spectral Data. Mathematics (2022).