Reproducing Kernel Functions: A general framework for Discrete Variable Representation

Since its introduction, the Discrete Variable Representation (DVR) basis set has become an invaluable representation of state vectors and Hermitian operators in non-relativistic quantum dynamics and spectroscopy calculations. On the other hand reproducing kernel (positive definite) functions have been widely employed for a long time to a wide variety of disciplines: detection and estimation problems in signal processing; data analysis in statistics; generating observational models in machine learning; solving inverse problems in geophysics and tomography in general; and in quantum mechanics. In this article it was demonstrated that, starting with the axiomatic definition of DVR provided by Littlejohn [1], it is possible to show that the space upon which the projection operator, defined in ref [1], projects is a Reproducing Kernel Hilbert Space (RKHS) whose associated reproducing kernel function can be used to generate DVR points and their corresponding DVR functions on any domain manifold (curved or not). It is illustrated how, with this idea, one may be able to `neatly’ address the long-standing challenge of building multidimensional DVR basis functions defined on curved manifolds.

💡 Research Summary

The paper establishes a deep connection between the Discrete Variable Representation (DVR) widely used in non‑relativistic quantum dynamics and the theory of reproducing kernel Hilbert spaces (RKHS). Starting from the axiomatic definition of DVR given by Littlejohn et al., the author shows that the subspace onto which the projection operator (P) maps is in fact an RKHS whose reproducing kernel can be employed to generate DVR points and the associated DVR basis functions on any manifold, whether flat or curved.

In the traditional DVR framework, one selects a finite orthonormal set ({\psi_j}{j=1}^N) in a Hilbert space (H), constructs the matrix representation of the position operators, and obtains eigen‑pairs that define the DVR grid ({x\alpha}) and the DVR functions (\Delta_\alpha(x)=\sum_{j=1}^N\psi_j(x)\psi_j(x_\alpha)). The paper rewrites this construction in kernel language: the kernel \