Solubility of boundary-value problems for a class of third-order operator-differential equations in a weighted space

Abstract

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.

On a Class of Operator-Differential Equations of the Third Order With Multiple Characteristics on the Whole Axis in the Weighted Space

“…We shall begin from studying the operator P (w) 0 acting from the space W 3 2,κ (R; H) into the space L 2,κ (R; H) in the following way:…”

“…A. Dubinskii [9], A. A. Shkalikov [22], S. S. Mirzoev [18], A. R. Aliev [1]- [4], and for the operator-differential equations of the fourth order with multiple characteristics are studied, for example, in the paper by A. R. Aliev [5]. It should be noted that widely enough matters of solvability of operator-differential equations are studied in spaces without weight (see the books by S. G. Krein [16], J.-L. Lions, and E. Magenes [17], V. I. Gorbachuk, and M. L. Gorbachuk [14], J.…”

On the solvability in a weight space of a third-order operator-differential equation with multiple characteristic

“…Such equations may arise, e.g., in modeling the dynamics of arcs and rings [2]. Among related works we mention [3][4][5][6][7][8]. In [3], the normal solvability of boundary value problems on a half axis for a certain class of higher order operator differential equations in the Sobolev-Slobodetskii space with exponential weight was proved.…”

“…A detailed study of the Fredholm solvability in a Hilbert space with exponential weight of boundary value problems on a half axis and on a finite interval for elliptic equa tions of any order with noncommuting operator coef ficients was performed in [4]. In [5][6][7], boundary value problems on a half axis for operator differential equations of orders 2 and 3 having simple characteris tics were considered. In [8], conditions for the regular solvability in the weight space of the boundary value problem for a class of fourth order operator differen tial equations with multiple characteristic were obtained.…”

On Solvability of Third-Order Operator Differential Equation with Parabolic Principal Part in Weighted Space

“…Despite the number of works covering the solvability of operator differential equations in weighted spaces being relatively small (see, e.g., Dubinskiȋ [3], Shkalikov [26], Mirzoev [27], and Aliev [28]), recently a lot of works appeared dedicated to the issues of regular and normal solvability in weighted spaces for operator differential equations with multiple characteristics (see Aliev [29], Mirzoev and Humbataliev [30], Aliev and Elbably [31], and Aliev and Lachinova [32]). But the equations considered in the abovementioned works, except for [32], belong to the class of quasielliptic operator differential equations [3].…”