Stability of spectral characteristics of boundary value problems for 2 × 2 Dirac type systems. Applications to the damped string

The paper is concerned with the stability property under perturbation Q→Q˜ of different spectral characteristics of a boundary value problem associated in L2([0,1];C2) with the following 2×2 Dirac type equation LU(Q)y=−iB−1y′+Q(x)y=λy,B=(b100b2),b1<0<b2,y=col(y1,y2), with a potential matrix Q∈Lp([0,1];C2×2) and subject to the regular boundary conditions Uy:={U1,U2}y=0. If b2=−b1=1 this equation is equivalent to one dimensional Dirac equation. Our approach to the spectral stability relies on the existence of the triangular transformation operators for system (0.1) with Q∈L1, which was established in our previous works. The starting point of our investigation is the Lipshitz property of the mapping Q→KQ±, where KQ± are the kernels of transformation operators for system (0.1). Namely, we prove the following uniform estimate: ‖KQ±−KQ˜±‖X∞,p2+‖KQ±−KQ˜±‖X1,p2⩽C⋅‖Q−Q˜‖p,Q,Q˜∈Up,r2×2,p∈[1,∞], on balls Up,r2×2 in Lp([0,1];C2×2). It is new even for Q˜=0. Here X∞,p2, X1,p2 are the special Banach spaces naturally arising in such problems. We also obtained similar estimates for Fourier transforms of KQ±. Both of these estimates are of independent interest and play a crucial role in the proofs of all spectral stability results discussed in the paper. For instance, as an immediate consequence of these estimates we get the Lipshitz property of the mapping Q→ΦQ(⋅,λ), where ΦQ(x,λ) is the fundamental matrix of the system (0.1). Assuming the spectrum ΛQ={λQ,n}n∈Z of LU(Q) to be asymptotically simple, denote by FQ={fQ,n}|n|>N a sequence of corresponding normalized eigenvectors, LU(Q)fQ,n=λQ,nfQ,n. Assuming boundary conditions (BC) to be strictly regular, we show that the mapping Q→ΛQ−Λ0 sends Lp([0,1];C2×2) either into ℓp′ or into the weighted ℓp-space ℓp({(1+|n|)p−2}); we also establish its Lipshitz property on compact sets in Lp([0,1];C2×2), p∈[1,2]. The proof of the second estimate involves as an important ingredient inequality that generalizes classical Hardy-Littlewood inequality for Fourier coefficients. It is also shown that the mapping Q→FQ−F0 sends Lp([0,1];C2×2) into the space ℓp′(Z;C([0,1];C2) of sequences of continuous vector-functions, and has the Lipshitz property on compacts sets in Lp([0,1];C2×2), p∈[1,2]. Certain modifications of these spectral stability results are also proved for balls Up,r2×2 in Lp([0,1];C2×2), p∈[1,2]. Note also that the proof of the Lipshitz property of the mapping Q→FQ−F0 involves the deep Carleson-Hunt theorem for maximal Fourier transform, while the proof of this property for the mapping Q→ΛQ−Λ0 relies on the estimates of the classical Fourier transform and is elementary in character. We apply our previous results to the damped string equation to establish the Riesz basis property and the asymptotic behavior of the eigenvalues of the corresponding dynamic generator under the assumptions d∈L1[0,ℓ], ρ∈W1,1[0,ℓ] on the damping coefficient and the density of the string, that are weaker than previously treated in the literature. We also establish Lipshitz dependence on d and ρ in ℓp-spaces of the remainders in the asymptotic formula for the eigenvalues. © 2022 Elsevier Inc.