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Comparative analysis of selected models of semiconductor superlattices

Mączka M., Pawłowski S., Plewako J.

Przegląd Elektrotechniczny

2011

R. 87, nr 8

93-95

Electrical properties of semiconductor lasers may be described with the help of the numerical calculations performed for the superlattices with the infinite and finite dimensions. In the paper two models of superlattices are compared. In the first approach the finite length of the structure is considered and the Schrödinger equation for the nonpolarized and polarized structures is solved. In the second approach delocalized Bloch functions over the whole structure of the superlattice are considered.

W pracy przedstawiono wybrane właściwości elektryczne laserów półprzewodnikowych, modelowanych za pomocą skończonych i nieskończonych półprzewodnikowych supersieci. Zamieszczono rozwiązania równań Schrödingera dla spolaryzowanej i niespolaryzowanej struktury supersieci.

Inverse/observability estimates for Schroedinger equations with variable coefficients

Triggiani R., Yao P.

Control and Cybernetics

1999

Vol. 28, no 3

627-664

We consider a general Schroedinger equation defined on an open bounded domain [Omega is a subset of R^n] with variable coefficients in both the elliptic principal part and in the first-order terms as well. At first, no boundary conditions (B.C.) are imposed. Our main result (Theorem 3.5) is a reconstruction, or inverse, estimate for solutions w: under checkable conditions on the coefficients of the principal part, the H[sup l](Omega)-energy at time t = T, or at time t = 0, is dominated by the L[sub2](Sigma)-norms of the boundary traces [...] and w[sub t] modulo an interior lower-order term. Once homogeneous B.C. are imposed, our results yield - under a uniqueness theorem, needed to absorb the lower order term - continuous observability estimates for both the Dirichlet and Neumann case, with an arbitrarily short observability time ; hence, by duality, exact controllability results. Moreover, no artificial geometrical conditions are imposed on the controlled part of the boundary in the Neuman case. In contrast to existing literature, the first step of our method employs a Riemann geometry approach to reduce the original variable coefficient principal part problem in [Omega is a subset of R^n] to a problem on an appropriate Riemannian manifold (determined by the coefficients of the principal part), where the principal part is the Laplacian. In our second step, we employ explicit Carleman estimates at the differential level to take care of the va.riable first-order (energy level) terms. In our third step, we employ micro-local analysis yielding a sharp trace estimate to remove artificial geometrical conditions on the controlled part of the boundary in the Neumann case.