Property C for ODE and Applications to an Inverse Problem for a Heat Equation

A. G. Ramm. "Property C for ODE and Applications to an Inverse Problem for a Heat Equation." Bulletin of the Polish Academy of Sciences. Mathematics 57.3 (2009): 243-249. <http://eudml.org/doc/281153>.

@article{A2009,
abstract = {Let $ℓ_j:= -d²/dx² + k²q_j(x)$, k = const > 0, j = 1,2, $0 < ess inf q_j(x) ≤ ess sup q_j(x) < ∞$. Suppose that (*) $∫_\{0\}^\{1\} p(x)u₁(x,k)u₂(x,k)dx = 0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_j$ solves the problem $ℓ_ju_j = 0$, 0 ≤ x ≤ 1, $u^\{\prime \}_j(0,k) = 0$, $u_j(0,k) = 1$. It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.},
author = {A. G. Ramm},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
language = {eng},
number = {3},
pages = {243-249},
title = {Property C for ODE and Applications to an Inverse Problem for a Heat Equation},
url = {http://eudml.org/doc/281153},
volume = {57},
year = {2009},
}

TY - JOUR
AU - A. G. Ramm
TI - Property C for ODE and Applications to an Inverse Problem for a Heat Equation
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2009
VL - 57
IS - 3
SP - 243
EP - 249
AB - Let $ℓ_j:= -d²/dx² + k²q_j(x)$, k = const > 0, j = 1,2, $0 < ess inf q_j(x) ≤ ess sup q_j(x) < ∞$. Suppose that (*) $∫_{0}^{1} p(x)u₁(x,k)u₂(x,k)dx = 0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_j$ solves the problem $ℓ_ju_j = 0$, 0 ≤ x ≤ 1, $u^{\prime }_j(0,k) = 0$, $u_j(0,k) = 1$. It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.
LA - eng
UR - http://eudml.org/doc/281153
ER -