Abstract
We consider the Dirac system on the interval [0, 1] with a spectral parameter mu in {mathbb {C}} and a complex-valued potential with entries from L_p[0,1], where 1le p. We study the asymptotic behavior of its solutions in a strip |mathrm{Im},mu |le d for mu rightarrow infty . These results allow us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm–Liouville operators associated with the aforementioned Dirac system.
Highlights
Consider a Cauchy problemD (x) + J(x)D(x) = AμD(x), D(0) = I, (1.1)where x ∈ [0, 1], Aμ = iμJ0, and J0 = J(x) = 0 σ1(x) σ2(x) 0 I := (1.2)μ ∈ C is a spectral parameter, and σj ∈ Lp[0, 1], j = 1, 2 are complex-valued functions where 1 ≤ p < 2
We study the asymptotic behavior of its solutions in a strip |Im μ| ≤ d for μ → ∞
These results allow us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm–Liouville operators associated with the aforementioned Dirac system
Summary
Μ ∈ C is a spectral parameter, and σj ∈ Lp[0, 1], j = 1, 2 are complex-valued functions where 1 ≤ p < 2. We study the asymptotic behavior of its solutions. The solution of (1.1) is a matrix D with entries from the space of absolutely continuous functions on [0, 1] (i.e. from the Sobolev space W11[0, 1]) satisfying (1.1) for a.e. x ∈ [0, 1]. In our case, this definition together with the equation imply that D has entries from Wp1[0, 1]
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