scholarly journals Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator

Entropy ◽  
2020 ◽  
Vol 22 (7) ◽  
pp. 722
Author(s):  
Stefan Klus ◽  
Feliks Nüske ◽  
Boumediene Hamzi
Keyword(s):  
Schrödinger Operator ◽  
Reproducing Kernel ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Schrodinger Operator ◽  
Koopman Operator ◽  
Reduction Techniques ◽  
Kernel Hilbert Spaces ◽  
Quantum Mechanical Systems ◽  
Matrix Eigenvalue Problems

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.

scholarly journals Reproducing kernel Hilbert spaces on manifolds: Sobolev and diffusion spaces

2020 ◽  
pp. 1-34
Author(s):  
Ernesto De Vito ◽  
Nicole Mücke ◽  
Lorenzo Rosasco
Keyword(s):  
Sobolev Spaces ◽  
Riemannian Manifolds ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Differential Operators ◽  
Reproducing Kernels ◽  
Reproducing Kernel Hilbert Spaces ◽  
Kernel Hilbert Spaces ◽  
Euclidean Setting ◽  
And Diffusion

We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples. While connections between Sobolev spaces, differential operators and RKHS are well known in the Euclidean setting, here we present a self-contained study of analogous connections for Riemannian manifolds. By collecting a number of results in unified a way, we think our study can be useful for researchers interested in the topic.

scholarly journals Computing the q-Numerical Range of Differential Operators

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Ahmed Muhammad ◽  
Faiza Abdullah Shareef
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Variational Methods ◽  
Schrödinger Operator ◽  
Orthonormal Basis ◽  
Differential Operators ◽  
Numerical Range ◽  
Stokes Operator ◽  
Schrodinger Operator ◽  
Operator Matrices

A linear operator on a Hilbert space may be approximated with finite matrices by choosing an orthonormal basis of thez Hilbert space. In this paper, we establish an approximation of the q-numerical range of bounded and unbounnded operator matrices by variational methods. Application to Schrödinger operator, Stokes operator, and Hain-Lüst operator is given.

scholarly journals Restricted interpolation by meromorphic inner functions

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Alexei Poltoratski ◽  
Rishika Rupam
Keyword(s):  
Half Plane ◽  
Schrödinger Operator ◽  
Differential Operators ◽  
Schrodinger Operator ◽  
Inner Functions ◽  
Spectral Problems ◽  
Upper Half Plane

AbstractMeromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent results concerning function theoretic properties of MIFs and show their connections with spectral problems for the Schrödinger operator.

scholarly journals Nontensorial generalised Hermite spectral methods for PDEs with fractional Laplacian and Schrodinger operators

2021 ◽  
Author(s):  
Changtao Sheng ◽  
Suna Ma ◽  
Huiyuan Li ◽  
Li-Lian Wang ◽  
Lueling Jia
Keyword(s):  
Schrödinger Operator ◽  
Spectral Methods ◽  
Fractional Laplacian ◽  
Eigenvalue Problems ◽  
Sparse Matrices ◽  
Hermite Polynomials ◽  
Accurate Solution ◽  
Hermite Functions ◽  
Inner Product ◽  
Schrodinger Operator

In this paper, we introduce  two families of  nontensorial  generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop  efficient and accurate spectral methods for solving  PDEs with integral fractional Laplacian (IFL) and/or  Schr\"{o}dinger operators in R^d. As a generalisation of the G. Szego's  family in 1D (1939),  the first family of  multivariate GHPs (resp. GHFs) are orthogonal with respect to the weight function |x|^{2\mu} e^{-|x|^2} (resp. |x|^{2\mu}) in R^d. We further  construct the adjoint generalised Hermite functions (A-GHFs), which have  an interwoven connection with the corresponding GHFs through  the Fourier transform,  and  are orthogonal with respect to the inner product [u,v]_{H^s(R^d)}=((-\Delta)^{s/2}u, (-\Delta)^{s/2} v)_{R^d} associated with the IFL of order s>0. As an immediate  consequence,  the spectral-Galerkin method using A-GHFs as basis functions  leads to a diagonal stiffness matrix for  the IFL (which is known to be notoriously difficult and expensive to discretise). The new basis also finds remarkably efficient  in solving  PDEs with the fractional  Schrodinger operator: (-\Delta)^s +|x|^{2\mu} with s\in (0,1] and \mu>-1/2 in R^d. We construct the second family of  multivariate nontensorial  Muntz-type GHFs, which are orthogonal with respect to an inner product associated with the underlying Schrodinger operator, and  are tailored to the singularity of the solution at the origin. We demonstrate that the Muntz-type GHF spectral method leads to sparse matrices and spectrally accurate solution to some  Schrodinger eigenvalue problems.

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