The paper is devoted to asymptotic behavior of the eigenvalues of the \(n \times n\) Dirac type equation \( y' + Q(x) y = i \lambda B(x) y, \quad y = \text {col}(y_1, \ldots , y_n), \quad x \in [0,\ell ], \) on a finite interval \([0,\ell ]\) subject to general regular boundary conditions \(C y(0) + D y(\ell ) = 1\) with \(C, D \in \mathbb {C}^{n \times n}\) . Here, \(Q = (Q_{jk})_{j,k=1}^n \in {L^{1}}({[0,\ell ]}; \mathbb {C}^{n \times n})\) is a potential matrix and \( B = {{\,\textrm{diag}\,}}(\beta _1, \ldots , \beta _n) = B^* \in L^1([0,\ell ];\mathbb {R}^{n \times n}) \) is a selfadjoint diagonal matrix “weight”. If \(n=2m\) and \(B(x) = {{\,\textrm{diag}\,}}(-I_m, I_m)\) , then this equation is equivalent to the \(n\times n\) Dirac equation. Under the assumption \({{\,\textrm{supp}\,}}(Q_{jk}) \subset {{\,\textrm{supp}\,}}(\beta _k - \beta _j)\) , it is proved that the deviation of the characteristic determinants \(\Delta _Q(\,\cdot \,)\) and \(\Delta _0(\,\cdot \,)\) of this boundary-value problem (BVP) and the unperturbed BVP (with \(Q \equiv 0\) ) is the Fourier transform of some integrable function, \( \Delta _Q(\lambda ) = \Delta _0(\lambda ) + \int \limits _{\widetilde{b}_-}^{\widetilde{b}_+} g(u) e^{i \lambda u} \, du, \quad g \in L^1[\widetilde{b}_-, \widetilde{b}_+]. \) This result is valid for arbitrary boundary conditions and arbitrary selfadjoint diagonal matrix function \(B(\,\cdot \,)\) . This representation is applied to study the distribution of zeros of the characteristic determinant \(\Delta _Q(\,\cdot \,)\) (the eigenvalues of the above BPV) and to show that \(\Delta _Q(\,\cdot \,)\) is always a function of a class A of exponential type, which is bounded on the real axis. Conditions guaranteeing that \(\Delta _Q(\,\cdot \,)\) is a sine-type function are found and sharp asymptotic formula for its zeros in this case is provided. Finally, it is proved that if the entries of the matrix \(B(\,\cdot \,)\) can change sign within the segment \([0,\ell ]\) , then, in general, even in the case of regular boundary conditions, the eigenvalues split into two branches: the “good” branch lies in the horizontal strip and is close to the eigenvalues of the unperturbed BVP, while the “bad” branch has nonzero density and imaginary parts that tend to infinifity. This effect is illustrated by a concrete \(2 \times 2\) example. Bibliography: 42 titles.
