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The local Borg-Marchenko uniqueness theorem for potentials locally smooth at the right endpoint. (English) Zbl 1485.34082

Summary: In this note we are concerned with the local Borg-Marchenko uniqueness theorem for potentials locally smooth at the right endpoint. We establish the so-called high-energy asymptotics of the difference \((m_{1}-m_{2})(z)\) of two Weyl-Titchmarsh functions \(m_{j}(z)\) corresponding to two Schrödinger operators \(H_j=-d^2/dx^2+q_j\), for \(j=1,2\) and \(q_1=q_2\) a.e. on [0, \(a\)], in \(L_{2}(0,b)\) with \(0<a<b\le \infty \), where the potentials \( q_j\) are sufficiently smooth in a right neighbourhood of the point \(a\) and their right derivatives at \(a\) coincide up to a certain order. Moreover, we also provide a new proof of the local Borg-Marchenko theorem.

MSC:

34A55 Inverse problems involving ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B20 Weyl theory and its generalizations for ordinary differential equations
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