scholarly journals Limiting absorption principle, generalized eigenfunctions, and scattering matrix for Laplace operators with boundary conditions on hypersurfaces

2018 ◽  
Vol 8 (4) ◽  
pp. 1443-1486 ◽  
Author(s):  
Andrea Mantile ◽  
Andrea Posilicano ◽  
Mourad Sini
Keyword(s):  
Boundary Conditions ◽  
Scattering Matrix ◽  
Laplace Operators ◽  
Limiting Absorption Principle ◽  
Generalized Eigenfunctions

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

scholarly journals Memory Efficient Method for Electromagnetic Multi-Region Models Using Scattering Matrices

2018 ◽  
Vol 23 (4) ◽  
pp. 71 ◽  
Author(s):  
C. Custers ◽  
J. Jansen ◽  
E. Lomonova
Keyword(s):  
Electromagnetic Field ◽  
Boundary Conditions ◽  
Efficient Method ◽  
Maximum Size ◽  
Scattering Matrix ◽  
Matrix Approach ◽  
Matrix Formulation ◽  
Scattering Matrices ◽  
Memory Efficient

This paper describes the scattering matrix approach to obtain the solution to electromagnetic field quantities in harmonic multi-layer models. Using this approach, the boundary conditions are solved in such way that the maximum size of any matrix used during the computations is independent of the number of regions defined in the problem. As a result, the method is more memory efficient than classical methods used to solve the boundary conditions. Because electromagnetic sources can be located inside the regions of a configuration, the scattering matrix formulation is developed to incorporate these sources into the solving process. The method is applied to a 3D electromagnetic configuration for verification.

scholarly journals Generalized eigenfunctions for quantum walks via path counting approach

2021 ◽  
pp. 2150019
Author(s):  
Takashi Komatsu ◽  
Norio Konno ◽  
Hisashi Morioka ◽  
Etsuo Segawa
Keyword(s):  
Asymptotic Behavior ◽  
Quantum Walk ◽  
Scattering Matrix ◽  
Quantum Walks ◽  
Tunneling Effect ◽  
Counting Approach ◽  
Finite Rank Perturbation ◽  
Generalized Eigenfunctions ◽  
The Green Function ◽  
Path Counting

We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at the spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial construction of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.

scholarly journals Limiting absorption principle and scattering matrix for Dirac operators with δ-shell interactions

2020 ◽  
Vol 61 (3) ◽  
pp. 033504
Author(s):  
Jussi Behrndt ◽  
Markus Holzmann ◽  
Andrea Mantile ◽  
Andrea Posilicano
Keyword(s):  
Dirac Operators ◽  
Scattering Matrix ◽  
Limiting Absorption Principle

scholarly journals Further results for a one-dimensional linear thermoelastic equation with Dirichlet-Dirichlet boundary conditions

2002 ◽  
Vol 43 (3) ◽  
pp. 449-462 ◽  
Author(s):  
Bao Zhu Guo
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Riesz Basis ◽  
Dirichlet Boundary ◽  
The State ◽  
Dirichlet Boundary Conditions ◽  
One Dimensional ◽  
Parallel Result ◽  
Generalized Eigenfunctions ◽  
Thermoelastic Equation

AbstractWe show that a sequence of generalized eigenfunctions of a one-dimensional linear thermoelastic system with Dirichiet-Dirichlet boundary conditions forms a Riesz basis for the state Hilbert space. This develops a parallel result for the same system with Dirichlet-Neumann or Neumann-Dirichlet boundary conditions.

scholarly journals Dissipative Dirac operator with general boundary conditions on time scales

2020 ◽  
Vol 72 (5) ◽  
Author(s):  
B. P. Allahverdiev ◽  
H. Tuna
Keyword(s):  
Dirac Operator ◽  
Boundary Conditions ◽  
Time Scales ◽  
Functional Model ◽  
Scattering Matrix ◽  
The Other ◽  
General Boundary ◽  
Dissipative Operator ◽  
General Boundary Conditions ◽  
Symmetric Operators

UDC 517.9 In this paper, we consider the symmetric Dirac operator on bounded time scales. With general boundary conditions, we describe extensions (dissipative, accumulative, self-adjoint and the other) of such symmetric operators. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.

scholarly journals R-function Theory for Bending Problem of Shallow Spherical Shells with Polygonal Boundary

2020 ◽  
Vol 6 (3) ◽  
pp. 512-522
Author(s):  
Shanqing Li ◽  
Hong Yuan ◽  
Xiongfei Yang ◽  
Huanliang Zhang ◽  
Qifeng Peng
Keyword(s):  
Boundary Conditions ◽  
Integral Equations ◽  
Green's Function ◽  
Function Theory ◽  
Winkler Foundation ◽  
Boundary Equation ◽  
Spherical Shells ◽  
Laplace Operators ◽  
Independent Equation ◽  
Polygonal Boundary

The governing differential equations of the bending problem of simply supported shallow spherical shells on Winkler foundation are simplified to an independent equation of radial deflection. The independent equation of radial deflection is decomposed to two Laplace operators by intermediate variable. The R-function theory is applied to describe a shallow spherical shell on Winkler foundation with concave boundary, and then a quasi-Green’s function is established by using the fundamental solution and the normalized boundary equation. The quasi-Green’s function satisfies the homogeneous boundary condition of the problem. The Laplace operators of the problem are reduced to two simultaneous Fredholm integral equations of the second kind by the Green’s formula. The singularity of the kernel of the integral equation is eliminated by choosing a suitable form of the normalized boundary equation. The integral equations are discretized into the homogeneous linear algebraic equations to proceed numerical computing. The singular term in the discrete equation is eliminated by the integral method. Some numerical examples are given to verify the validity of the proposed method in calculating simple boundary conditions and polygonal boundary conditions. A comparison with the ANSYS finite element (FEM) solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present method.

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