On the formula for characteristic determinants of boundary value problems for n × n Dirac type systems and its applications

• Published in: Advances in mathematics (New York. 1965) Vol. 478; p. 110389
• Main Authors: Lunyov, Anton A., Malamud, Mark M.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.10.2025
• Subjects: Eigenvalues asymptotic; Regular boundary conditions; Systems of ordinary differential equations; 34L40; 47E05; Systems of ordinary differential equations; Regular boundary conditions; Eigenvalues asymptotic; 35L35; 34L10; 34L20
• ISSN: 0001-8708
• DOI: 10.1016/j.aim.2025.110389

The paper is concerned with the spectral properties of the boundary value problems (BVP) associated with the following n×n Dirac type equation:−iy′−iQ(x)y=λB(x)y,y=col(y1,…,yn),x∈[0,ℓ], on a finite interval [0,ℓ] subject to the general two-point boundary conditions Cy(0)+Dy(ℓ)=0 with C,D∈Cn×n. Here Q=(Qjk)j,k=1n is an integrable potential matrix and B=diag(β1,…,βn)=B⁎ is a diagonal integrable matrix “weight”. If n=2m and B(⋅)=diag(−Im,Im), this equation turns into n×n Dirac equation. First, assuming that supp(Qjk)⊂supp(βk−βj), we show that the deviation ΦQ(⋅,λ)−Φ0(⋅,λ) of the fundamental matrix solutions to the above perturbed and unperturbed (Q=0) equation is represented as a Fourier transform of a certain matrix kernel KQ(⋅,⋅) from a special Banach space. This result is used to prove our main result, the following formula for the deviation of the characteristic determinants ΔQ(⋅) and Δ0(⋅) of two (perturbed and unperturbed) BVPs as a Fourier transform,ΔQ(λ)=Δ0(λ)+∫b−b+g(u)eiλudu,g∈L1[b−,b+], where b± are explicitly expressed via entries of the matrix function B(⋅). In turn, assuming that each function βk(⋅) is of fixed sign (so-called “fixed sign” condition), this representation yields asymptotic behavior of the spectrum in the case of regular boundary conditions. Namely, we show that λm=λm0+o(1) as m→∞, where {λm}m∈Z and {λm0}m∈Z are sequences of eigenvalues of perturbed and unperturbed BVP, respectively. It is also shown that for Q∈Lp, p∈(1,2], the following estimate holds under additional condition on B(⋅):∑m∈Z|λm−λm0|p′+∑m∈Z(1+|m|)p−2|λm−λm0|p<∞,p′:=p/(p−1). In the case of Dirac operator, we show that the sequence of the eigenvalues splits into the union of n branches asymptotically close to arithmetic progressions {2πk−iLog(−μs)}k∈Z, s∈{1,…,n}, where μ1,…,μn are the eigenvalues of a certain n×n matrix explicitly constructed via the matrices C and D.