Schrödinger Operator Theory in Function Spaces

Schrödinger operator theory in function spaces examines the mapping properties and spectral behaviour of operators of the form L = –Δ + V, where Δ denotes the Laplace operator and V a nonnegative potential. Central to this field is the interplay between the analytical structure imposed by V—often assumed to satisfy a reverse Hölder inequality—and the choice of function space on which L acts. Over recent decades, researchers have developed Hardy, Morrey, BMO and Lipschitz spaces adapted to L, enabling precise control of singular integrals, heat semigroups and fractional powers of the operator. These advances underpin applications to partial differential equations, quantum mechanics and geometric analysis, offering robust tools to address boundary‐value problems, regularity of solutions and decay estimates. Key themes include the characterisation of boundary behaviour via Carleson measures, boundedness of Riesz and Littlewood–Paley operators, and the study of fractional semigroups that generalise classical heat kernels. The theory not only unifies disparate techniques from harmonic analysis and spectral theory but also reveals deep connections between functional calculus and potential theory in both Euclidean and non‐Euclidean settings.

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Schrödinger Operator Theory in Function Spaces publication trend

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Technical terms

Schrödinger operator: A second‐order differential operator of the form –Δ + V, combining the Laplacian and a potential term.

Reverse Hölder class: A family of weight functions satisfying an integral inequality that bounds averages of powers by powers of averages, controlling local oscillation of the potential.

Heat semigroup: The family {e^–tL}_t>0 of operators solving the heat equation ∂_tu + Lu = 0, encoding time‐evolution under L.

Morrey spaces: Function spaces refining Lebesgue norms by measuring local integrability over balls with a scaling exponent, suited to operators with nonuniform behaviour.

Lipschitz spaces: Spaces of functions whose increments satisfy a Hölder condition of specified order, generalised in this context to operator‐adapted definitions.

Fractional integral operator: An operator generalising classical potential kernels, realised as fractional powers or resolvent integrals of L, and mapping between different function spaces.

Carleson measure: A measure on a domain crossed with a positive parameter (often time or scale) satisfying a growth condition that quantifies boundary regularity of solutions.

References

1. H p spaces associated with Schrödinger operators with potentials from reverse Hölder classes. Colloquium Mathematicum (2003).
2. Regularity of Fractional Heat Semigroup Associated with Schrödinger Operators. Fractal and Fractional (2022).
3. Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators. Mathematics (2022).
4. Boundedness of second-order Riesz transforms on weighted Hardy and $BMO$ spaces associated with Schrödinger operators. Comptes Rendus Mathématique (2021).
5. Fractional integral associated with the Schrödinger operators on variable exponent space. Electronic Research Archive (2023).
6. End Point Estimate of Littlewood‐Paley Operator Associated to the Generalized Schrödinger Operator. Journal of Function Spaces (2021).