scholarly journals Scattering analysis and spectrum of discrete Schrödinger equations with transmission conditions

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5391-5399 ◽  
Author(s):  
Elgiz Bairamov ◽  
Yelda Aygar ◽  
Dilara Karslıoğlu
Keyword(s):  
Boundary Value Problem ◽  
Discrete Spectrum ◽  
Boundary Value ◽  
Transmission Conditions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Scattering Function ◽  
Dinger Equation ◽  
Special Case ◽  
Jost Solution

In this paper, we present an investigation about scattering analysis of an transmission boundary value problem (TBVP) which consists a discrete Schr?dinger equation and transmission conditions. Discussing the Jost solution and scattering function of this problem, we find the properties of scattering function of this problem by using the scattering solutions. We also investigate the discrete spectrum of this boundary value problem. Furthermore, we apply the results on an example which is the special case of main TBVP and we discuss the existence of eigenvalues of this example.

scholarly journals Stabilization of higher order Schrödinger equations on a finite interval: Part II

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Türker Özsarı ◽  
Kemal Cem Yılmaz
Keyword(s):  
Boundary Value Problem ◽  
Finite Interval ◽  
Boundary Value ◽  
Initial Boundary ◽  
Higher Order ◽  
Observer Design ◽  
Target System ◽  
Schrodinger Equations ◽  
Left Endpoint ◽  
Schrödinger Equations

<p style='text-indent:20px;'>Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [<xref ref-type="bibr" rid="b3">3</xref>] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.</p>

scholarly journals Discrete Sturm-Liouville equations with point interaction

2022 ◽  
Author(s):  
Güher Özbey ◽  
yelda AYGAR ◽  
Basak Oznur
Keyword(s):  
Boundary Value Problem ◽  
Discrete Spectrum ◽  
Resolvent Operator ◽  
Boundary Value ◽  
Point Interaction ◽  
Scattering Function ◽  
Asymptotic Equation ◽  
Sturm Liouville

Scattering solutions and several properties of scattering function of a discrete Sturm-Liouville boundary value problem with point interaction (PBVP) are derived. Moreover, resolvent operator, continuous and discrete spectrum of this PBVP are investigated. An asymptotic equation is utilized to get the properties of eigenvalues. An example illustrating the main results is given.

scholarly journals STOCHASTIC EVOLUTION AS A QUASICLASSICAL LIMIT OF A BOUNDARY VALUE PROBLEM FOR SCHRÖDINGER EQUATIONS

2002 ◽  
Vol 05 (01) ◽  
pp. 61-91
Author(s):  
V. P. BELAVKIN ◽  
V. N. KOLOKOL'TSOV
Keyword(s):  
Boundary Value Problem ◽  
Semiclassical Limit ◽  
Boundary Value ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fock Spaces ◽  
Stochastic Evolution ◽  
Quasiclassical Limit ◽  
Continuous Quantum Measurements ◽  
Bounded Below

We develop systematically a new unifying approach to the analysis of linear stochastic, quantum stochastic and even deterministic equations in Banach spaces. Solutions to a wide class of these equations (in particular those describing the processes of continuous quantum measurements) are proved to coincide with the interaction representations of the solutions to certain Dirac type equations with boundary conditions in pseudo-Fock spaces. The latter are presented as the semiclassical limit of an appropriately dressed unitary evolutions corresponding to a boundary-value problem for rather general Schrödinger equations with bounded below Hamiltonians.

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