Spectral Theory of Pseudodifferential Operators on Manifolds
Summary
The spectral theory of pseudodifferential operators on manifolds explores how the geometry and topology of a smooth manifold govern the distribution of eigenvalues and the structure of eigenfunctions of generalised differential operators. Central to this field is the symbol calculus, which encodes operator behaviour via principal and subprincipal symbols on the cotangent bundle. Semiclassical and microlocal methods establish Weyl’s law and its refinements, linking the asymptotic growth of the eigenvalue counting function to the volume of phase space and boundary contributions. Heat-kernel and resolvent expansions yield spectral invariants that appear in index theorems and determine geometric quantities such as volume, curvature integrals and characteristic classes. Recent advances extend classical results to operators with non-smooth coefficients, fractional-order elliptic operators and settings with limited boundary regularity, broadening applications in quantum mechanics, inverse problems and continuum mechanics. Developments in almost-invariant subspace decompositions and refined spectral clustering have illuminated the propagation of singularities in hyperbolic systems. Overall, the interplay between global analysis, microlocal techniques and operator algebras continues to deepen our understanding of how manifold structure is encoded in spectral data, with practical outcomes ranging from spectral geometry to computational spectral methods.
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Spectral Theory of Pseudodifferential Operators on Manifolds publication trend
The graph below shows the total number of articles in spectral theory of pseudodifferential operators on manifolds across all publications each year (not limited to Nature Index journals).
Technical terms
Pseudodifferential operator: A linear operator extending differential operators by allowing symbols with non-polynomial dependence on cotangent variables, defined via oscillatory integrals.
Principal symbol: The highest-order component of a pseudodifferential symbol, determining ellipticity and the leading term in spectral asymptotics.
Weyl’s law: An asymptotic formula for the counting function of eigenvalues of elliptic operators, relating spectral growth to the volume of phase space.
Resolvent: The operator (P–λI)⁻¹ defined for λ outside the spectrum, whose trace expansions yield heat-kernel coefficients and spectral invariants.
Spectral invariant: A geometric or topological quantity encoded in spectral data, often appearing as coefficients in heat-kernel or resolvent expansions.
References
- 100 years of Weyl’s law. Bulletin of Mathematical Sciences (2016).
- Invariant subspaces of elliptic systems II: Spectral theory. Journal of Spectral Theory (2022).
- Truncation quantization in the edge calculus. Journal of Pseudo-Differential Operators and Applications (2023).
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