On an inverse problem for a second-order differential equation

Abstract

In this work the exchange interactions in V15 magnetic molecular nanoclusters are examined. The magnetization reversal process is investigated at various values of the exchange constant set by means of the Monte Carlo method. It is shown that the most exact agreement with experimental results of the field dependence of susceptibility is observed in the following set of exchange interaction constants in the V15 magnetic molecular nanocluster: J = 500 K, J = 150 K, J = 225 K, J1 = 50 K, J2 = 50 K. For the first time, the heat capacity shows two maxima located in close proximity of each of the three transitions from the low-spin state into the high-spin behavior.

Non-self-adjoint boundary-value problem with discontinuous density function

“…In particular, Gasymov [4] considered the SLP (1)-(2) with a real-valued potential q to show that it has only a finite number of simple eigenvalues and that the half-axis constitutes its absolutely continuous spectrum. Afterwards, in [8] Guseinov and Pashaev handled the same BVP to investigate the inverse scattering problem. In [10], Darwish brought a non-self-adjoint SLP with discontinuous density function under investigation and gave two examples to draw the attention to the difficulty in the study of distribution and finiteness of eigenvalues.…”

“…We denote by S(x, k) and C(x, k) the solutions of Equation (1) satisfying (8). S(x, k) and C(x, k) have the following representations:…”

“…In addition, the existence of discontinuous coefficient (3) in Equation (1) makes the investigation of the problem quite challenging as it strongly influences the structure of Jost solution and hence the treatment of the problem. For instance, while the Jost solution is given by f (x, k) = e ikx + ∞ x K(x, t)e ikt dt in continuous case (see [7, Appendix II, Theorem 2]), the Jost solution in discontinuous case (see [8,Theorem 1]) is obtained as…”

“…(x) ≡ 1, spectral analysis of the non-self-adjoint BVP (1)- (2) is intensively investigated in [7]. In the case 0< = 1, the inverse problem corresponding to BVP (1)- (2) is handled in [4,8], in which q is assumed to be real-valued function. Note that q is allowed to take complex values in this study.…”

Integral Representations for the Solutions of the Generalized Schroedinger Equation in a Finite Interval

“…Inverse scattering problem for Equation (1) with ( ) 0 p x = on the half line [ ) 0, +∞ was investigated and the complete solution of this problem was given in [38] where the new integral representation, similar to transformation operators [9], was obtained for the Jost solution of the discontinuous Sturm-Liouville equation. Direct and inverse scattering problems on the half-line for the discontinuous Sturm-Liouville equation with eigenparameter dependent boundary conditions have been investigated in [41].…”