Minimal potential results for Schrödinger equations with Neumann boundary conditions

2017 ◽  
Vol 29 (6) ◽  
Author(s):  
Julian Edward ◽  
Steve Hudson ◽  
Mark Leckband
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Schrodinger Equations ◽  
Schrödinger Equations

AbstractWe consider the boundary value problem

scholarly journals Non-parabolic interface motion for the one-dimensional Stefan problem: Neumann boundary conditions

2017 ◽  
Vol 21 (6 Part B) ◽  
pp. 2699-2708 ◽  
Author(s):  
Jose Otero ◽  
Ernesto Hernandez ◽  
Ruben Santiago ◽  
Raul Martinez ◽  
Francisco Castillo ◽  
...  
Keyword(s):  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Large Time ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Interface Position

In this work, we study the liquid-solid interface dynamics for large time intervals on a 1-D sample, with homogeneous Neumann boundary conditions. In this kind of boundary value problem, we are able to make new predictions about the interface position by using conservation of energy. These predictions are confirmed through the heat balance integral method of Goodman and a generalized non-classical finite difference scheme. Since Neumann boundary conditions imply that the specimen is thermally isolated, through well stablished thermodynamics, we show that the interface behavior is not parabolic, and some examples are built with a novel interface dynamics that is not found in the literature. Also, it is shown that, on a Neumann boundary value problem, the position of the interface at thermodynamic equilibrium depends entirely on the initial temperature profile. The prediction of the interface position for large time values makes possible to fine tune the numerical methods, and given that energy conservation demands highly precise solutions, we found that it was necessary to develop a general non-classical finite difference scheme where a non-homogeneous moving mesh is considered. Numerical examples are shown to test these predictions and finally, we study the phase transition on a thermally isolated sample with a liquid and a solid phase in aluminum.

Recovering differential operators with two constant delays under Dirichlet/Neumann boundary conditions

2020 ◽  
Vol 28 (2) ◽  
pp. 237-241
Author(s):  
Biljana M. Vojvodic ◽  
Vladimir M. Vladicic
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions

AbstractThis paper deals with non-self-adjoint differential operators with two constant delays generated by {-y^{\prime\prime}+q_{1}(x)y(x-\tau_{1})+(-1)^{i}q_{2}(x)y(x-\tau_{2})}, where {\frac{\pi}{3}\leq\tau_{2}<\frac{\pi}{2}<2\tau_{2}\leq\tau_{1}<\pi} and potentials {q_{j}} are real-valued functions, {q_{j}\in L^{2}[0,\pi]}. We will prove that the delays and the potentials are uniquely determined from the spectra of four boundary value problems: two of them under boundary conditions {y(0)=y(\pi)=0} and the remaining two under boundary conditions {y(0)=y^{\prime}(\pi)=0}.

scholarly journals Gradient estimates via rearrangements for solutions of some Schrödinger equations

2018 ◽  
Vol 16 (03) ◽  
pp. 339-361 ◽  
Author(s):  
Sibei Yang ◽  
Der-Chen Chang ◽  
Dachun Yang ◽  
Zunwei Fu
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Gradient Estimates ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Elliptic Boundary Value ◽  
Neumann Boundary Value Problems ◽  
Laplace Type

In this paper, by applying the well-known method for dealing with [Formula: see text]-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schrödinger equations, under the weak regularity assumption on the boundary of domains. As applications, the gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained.

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 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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