scholarly journals Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

2010 ◽  
Vol 298 (2) ◽  
pp. 461-484
Author(s):  
Anna Maltsev
Keyword(s):  
Schrödinger Operator ◽  
Reproducing Kernel ◽  
Half Line ◽  
Schrodinger Operator ◽  
Universality Limits

EXISTENCE OF ALMOST EXPONENTIALLY DECAYING STATES FOR BARRIER POTENTIALS

2001 ◽  
Vol 13 (03) ◽  
pp. 267-305 ◽  
Author(s):  
RICHARD LAVINE
Keyword(s):  
Error Estimate ◽  
Half Plane ◽  
Imaginary Part ◽  
Schrödinger Operator ◽  
Small Error ◽  
Lower Half ◽  
Half Line ◽  
Schrodinger Operator ◽  
Lower Half Plane ◽  
Decaying States

For a Schrödinger operator H on the half line whose potential has a trapping barrier, and is convex outside the barrier, there exists a φ, supported mostly inside the barrier, such that for t>0, <φ, e-iHtφ>~e-izt up to a small error, where φ is obtained by cutting off a nonnormalizable solution ψ of Hψ=zψ, and z is in the lower half-plane. The imaginary part of z is estimated explicitly, and the error estimate is explicitly proportional to | Im z log | Im z‖.

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.

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