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  • 1.
    Aarao, J
    et al.
    University of S Australia, Australia .
    Miklavcic, Stan J.
    Linköping University, Department of Science and Technology. Linköping University, The Institute of Technology.
    Ward, D A.
    University of S Australia, Australia .
    Extended-domain-eigenfunction method (EDEM): a study of ill posedness and regularization2013In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 46, no 8Article in journal (Refereed)
    Abstract [en]

    The extended-domain-eigenfunction method (EDEM) proposed for solving elliptic boundary value problems on annular-like domains requires an inversion process. The procedure thus represents an ill-posed problem, whose numerical solution involves an ill-conditioned system of equations. In this paper, the ill-posed nature of EDEM is studied and numerical solutions based on regularization schemes are considered. It is shown that the EDEM solution methodology lends itself naturally to a formulation in terms of the well-known iterative Landweber method and the more general and faster converging semi-iterative regularization schemes. Theoretical details and numerical results of the regularization schemes are presented for the case of the two-dimensional Laplace operator on annular domains.

  • 2.
    Enblom, Alexandra
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Resolvent estimates and bounds on eigenvalues for Dirac operators on the half-line2018In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 51, no 16, article id 165203Article in journal (Refereed)
    Abstract [en]

    Estimates for the eigenvalues of non-self-adjoint 1D Dirac operators considered on the half-line are obtained in terms of the L p -norms of the potentials. The proofs are based on the resolvent estimates established for the free Dirac operator.

  • 3.
    Enblom, Alexandra
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Resolvent estimates and bounds on eigenvalues for Dirac operators on the half-line2018In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 51, no 16, article id 165203Article in journal (Refereed)
    Abstract [en]

    Estimates for the eigenvalues of non-self-adjoint 1D Dirac operators considered on the half-line are obtained in terms of the L-p-norms of the potentials. The proofs are based on the resolvent estimates established for the free Dirac operator.

  • 4.
    Jogenfors, Jonathan
    et al.
    Linköping University, Department of Electrical Engineering, Information Coding. Linköping University, The Institute of Technology.
    Larsson, Jan-Åke
    Linköping University, Department of Electrical Engineering, Information Coding. Linköping University, The Institute of Technology.
    Energy-time entanglement, elements of reality, and local realism2014In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 47, no 42, p. 424032-Article in journal (Refereed)
    Abstract [en]

    The Franson interferometer, proposed in 1989 (Franson 1989 Phys. Rev. Lett. 62 2205-08), beautifully shows the counter-intuitive nature of light. The quantum description predicts sinusoidal interference for specific outcomes of the experiment, and these predictions can be verified in experiment. In the spirit of Einstein, Podolsky, and Rosen it is possible to ask if the quantum-mechanical description (of this setup) can be considered complete. This question will be answered in detail in this paper, by delineating the quite complicated relation between energy-time entanglement experiments and Einstein-Podolsky-Rosen (EPR) elements of reality. The mentioned sinusoidal interference pattern is the same as that giving a violation in the usual Bell experiment. Even so, depending on the precise requirements made on the local realist model, this can imply (a) no violation, (b) smaller violation than usual, or (c) full violation of the appropriate statistical bound. Alternatives include (a) using only the measurement outcomes as EPR elements of reality, (b) using the emission time as EPR element of reality, (c) using path realism, or (d) using a modified setup. This paper discusses the nature of these alternatives and how to choose between them. The subtleties of this discussion needs to be taken into account when designing and setting up experiments intended to test local realism. Furthermore, these considerations are also important for quantum communication, for example in Bell-inequality-based quantum cryptography, especially when aiming for device independence.

  • 5.
    Larsson, Jan-Åke
    Linköping University, Department of Electrical Engineering, Information Coding. Linköping University, The Institute of Technology.
    Loopholes in Bell inequality tests of local realism2014In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 47, no 42, p. 424003-Article, review/survey (Refereed)
    Abstract [en]

    Bell inequalities are intended to show that local realist theories cannot describe the world. A local realist theory is one where physical properties are defined prior to and independent of measurement, and no physical influence can propagate faster than the speed of light. Quantum-mechanical predictions for certain experiments violate the Bell inequality while a local realist theory cannot, and this shows that a local realist theory cannot give those quantum-mechanical predictions. However, because of unexpected circumstances or loopholes in available experiment tests, local realist theories can reproduce the data from these experiments. This paper reviews such loopholes, what effect they have on Bell inequality tests, and how to avoid them in experiment. Avoiding all these simultaneously in one experiment, usually called a loophole-free or definitive Bell test, remains an open task, but is very important for technological tasks such as device-independent security of quantum cryptography, and ultimately for our understanding of the world.

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