Reconstruction of the potential from I-function

DOI10.1515/JIIP.2002.10.4.385zbMATH Open1020.34012arXivmath-ph/0102028OpenAlexW1987203324MaRDI QIDQ4794286

Alexander G. Ramm

Publication date: 12 October 2003

Published in: Journal of Inverse and Ill-posed Problems (Search for Journal in Brave)

Abstract: If $f(x,k)$ is the Jost solution and $f(x)=f(0,k)$, then the $I$-function is $I(k):=frac{f}^{p}rime(0,k)f(k)$. It is proved that $I(k)$ is in one-to-one correspondence with the scattering triple $mathcalS:=S(k),k_j,s_j,quad1leqjleqJ$ and with the spectral function $ho(lambda)$ of the Sturm-Liouville operator $l=-frac{d}^{2}d{x}^2+q(x)$ on $(0,infty)$ with the Dirichlet condition at $x=0$ and $$. Analytical methods are given for finding $mathcalS$ from $I(k)$ and $I(k)$ from $mathcalS$, and $ho(lambda)$ from $I(k)$ and $I(k)$ from $ho(lambda)$. Since the methods for finding $q(x)$ from $mathcalS$ or from $ho(lambda)$ are known, this yields the methods for finding $q(x)$ from $I(k)$.

Full work available at URL: https://arxiv.org/abs/math-ph/0102028

zbMATH Keywords

Sturm-Liouville operatorJost solution

Mathematics Subject Classification ID

Sturm-Liouville theory (34B24) Inverse problems involving ordinary differential equations (34A55)

Related Items (2)

$L^1$ convergence of the reconstruction formula for the potential function ⋮ Reconstruction of Potential Function for Diffusion Operator

Recommendations

• Unnamed Item 👍 👎
• Recovery of the potential from \(I\)-function 👍 👎
• Reconstruction of potential function for Sturm-Liouville operator with Coulomb potential 👍 👎
• Reconstructing potentials from zeros of one eigenfunction 👍 👎
• Reconstruction of Potential Function for Diffusion Operator 👍 👎
• Reconstruction of a Potential on the Line That is a Priori Known on the Half Line 👍 👎
• Reconstruction of a potential from the eigenvalues of various problems 👍 👎
• Reconstructing the Potential Function in a Formulation of Quantum Mechanics Based on Orthogonal Polynomials 👍 👎
• Reconstruction of the potential from partial Cauchy data for the Schroedinger equation 👍 👎
• Reconstruction formula for the potential function of Sturm–Liouville problem with eigenparameter boundary condition 👍 👎

This page was built for publication: Reconstruction of the potential from I-function