Vol 66, No 3 (2020): Spectral Analysis


Spectral Analysis of One-Dimensional Dirac System with Summable Potential and Sturm- Liouville Operator with Distribution Coefficients

Savchuk A.M., Sadovnichaya I.V.


We consider one-dimensional Dirac operator LP,U with Birkhoff regular boundary conditions and summable potential P(x) on [0, π]. We introduce strongly and weakly regular operators. In both cases, asymptotic formulas for eigenvalues are found. In these formulas, we obtain main asymptotic terms and estimates for the second term. We specify these estimates depending on the functional class of the potential: Lp[0,π] with p ∈ [1,2] and the Besov space Bp,p'θ[0,π] with p ∈ [1,2] and θ ∈ (0,1/p). Additionally, we prove that our estimates are uniform on balls Pp,θR Then we get asymptotic formulas for normalized eigenfunctions in the strongly regular case with the same residue estimates in uniform metric on x ∈ [0,π]. In the weakly regular case, the eigenvalues λ2n and λ2n+1 are asymptotically close and we obtain similar estimates for two-dimensional Riesz projectors. Next, we prove the Riesz basis property in the space (L2[0,π])2  for a system of eigenfunctions and associated functions of an arbitrary strongly regular operator LP,U. In case of weak regularity, the Riesz basicity of two-dimensional subspaces is proved.

In parallel with the LP,U operator, we consider the Sturm–Liouville operator Lq,U generated by the differential -y'' + q(x)y expressionwith distribution potential q of first-order singularity (i.e., we assume that the primitive u = q(1) belongs to L2[0, π]) and Birkhoff-regular boundary conditions. We reduce to this case -(τ1y')'+i(σy)'+iσy'+τ0y, operators of more general form where τ'1, σ,τ0(-1)L2  and τ1>0. For operator Lq,U, we get the same results on the asymptotics of eigenvalues, eigenfunctions, and basicity as for operator LP,U .

Then, for the Dirac operator LP,U, we prove that the Riesz basis constant is uniform over the ballsPp,θR for p>1 or θ>0. The problem of conditional basicity is naturally generalized to the problem

of equiconvergence of spectral decompositions in various metrics. We prove the result on equiconvergence by varying three indices: f Lμ[0,π] (decomposable function), PL[0,π] (potential), and Sm-Sm00, m in Lν[0,π]  (equiconvergence of spectral decompositions in the corresponding norm). In conclusion, we prove theorems on conditional and unconditional basicity of the system of eigenfunctions and associated functions of operator LP,U in the spaces Lμ[0,π], μ2, and in various Besov spaces Bp,qθ[0,π].

Contemporary Mathematics. Fundamental Directions. 2020;66(3):373-530
pages 373-530 views

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