In this note, we establish sufficient conditions for the correct and unique solvability of various boundary value problems for a class of even-order operator-differential equations on the half-axis. These conditions are unimprovable in terms of operator coefficients of the equation. We note that the principal part of the equation under study suffers a discontinuity.In this note, we establish sufficient conditions in terms of operator coefficients for correct and unique solvability of boundary value problems for a class of even-order operator-differential equations whose principal part suffers a discontinuity. Such problems have many applications, for example, in elasticity theory. In particular, some problems for multilayer bodies are simulated using equations with piecewise constant coefficients.Let A be a self-adjoint positive definite operator in a separable Hilbert space H . We consider the equationwhere f (t) ∈ L 2 (R + ; H) (see [1]) and ρ(t) = α for 0 t T and ρ(t) = β for T < t < +∞. Here α and β are positive and, generally, unequal numbers (for definiteness, α β) and A j (t), j = 1, . . . , 2k, are linear, generally unbounded, operators defined for almost all t ∈ R + .By W 2k 2 (R + ; H), we denote the space of H -valued functions u(t) such that u (2k) (t) ∈ L 2 (R + ; H) and A 2k u(t) ∈ L 2 (R + ; H) with the norm [2] H), we add to Eq. (1) boundary conditions at zero of the formwhere m i are positive integers such that 0 m 0 < m 1 < · · · < m k−1 2k−1 and m i +m j = 2k−1, 0 i, j 2k − 1.Definition. If a vector function u(t) ∈ W 2k 2 (R + ; H) satisfies Eq. (1) almost everywhere in R + and if the boundary conditions (2) hold in the sense thatthen u(t) is called a regular solution of the boundary value problem (1), (2). We set • W 2k 2 (R + ; H; {m i }) ≡ {u(t) : u(t) ∈ W 2k 2 (R + ; H), u (m i ) (0) = 0, i = 0, . . . , k − 1}

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The effect of Dzialoshinski-Moriya (DM) interaction on thermal entanglement of a two-qubit XXZ spin chain in a homogenous magnetic field is investigated. It is found that the DM interaction can enhance thermal entanglement. When D is large enough, the entanglement can exist for larger temperatures and strong magnetic field.