Shooting function for 1D Schrödinger operators

@article{MacKay2022ShootingFF,
  title={Shooting function for 1D Schr{\"o}dinger operators},
  author={R. S. MacKay},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2022},
  volume={55}
}
  • R. MacKay
  • Published 9 May 2022
  • Mathematics
  • Journal of Physics A: Mathematical and Theoretical
For Schrödinger operators with suitable 1D potentials, focussing particularly on those that go to infinity at infinity, a characteristic function is constructed, via shooting functions. It is proved to be entire and its zeroes to be the eigenvalues. 

References

SHOWING 1-10 OF 35 REFERENCES

The Shooting Method for Solving Eigenvalue Problems

  • Xi Chen
  • Mathematics, Computer Science
  • 1996
Abstract The shooting method is a numerically effective approach to solving certain eigenvalue problems, such as that arising from the Schrodinger equation for the two-dimensional hydrogen atom with

The Evans function and the Weyl-Titchmarsh function

We describe relations between the Evans function, a modern tool in the study of stability of traveling waves and other patterns for PDEs, and the classical Weyl-Titchmarsh function for singular

Functional determinants in quantum field theory

Functional determinants of differential operators play a prominent role in theoretical and mathematical physics, and in particular in quantum field theory. They are, however, difficult to compute in

Functional determinants for general Sturm–Liouville problems

Simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest. We extend the range of situations for which these hold to cover systems of

Oscillation theory and semibounded canonical systems

Oscillation theory locates the spectrum of a differential equation by counting the zeros of its solutions. We present a version of this theory for canonical systems $Ju'=-zHu$ and then use it to

The Schr\"odinger operator with Morse potential on the right half line

This paper studies the Schr\"odinger operator with Morse potential on a right half line [u, \infty) and determines the Weyl asymptotics of eigenvalues for constant boundary conditions. It obtains

Semiclassically weak reflections above analytic and non-analytic potential barriers

The coefficient r for reflection above a barrier V(x) is computed semiclassically (i.e. as h(cross) to 0) employing an exact multiple-reflection series whose mth term is a (2m+1)-fold integral. If

Towards a spectral proof of Riemann's hypothesis

The paper presents evidence that Riemann's xi function evaluated at 2 sqrt(E) could be the characteristic function P(E) for the magnetic Laplacian minus 85/16 on a surface of curvature -1 with

Eigenfunction Expansions associated with Second-Order Differential Equations

where for each r € HO, oo) A(r) is an operator in a Hilbert space H and & acts on ^-valued functions on fO, co). Restricting the domain of <£ appropriately, we can regard 3? as an operator in f) = £2

Canonical systems and de Branges spaces

This is an exposition of the inverse spectral theory of canonical systems based on de Branges spaces of entire functions