On a class of non-self-adjoint periodic boundary value problems with discrete real spectrum

In [arXiv:0801.0172] we examined a family of periodic Sturm-Liouville problems with boundary and interior singularities which are highly non-self-adjoint but have only real eigenvalues. We now establish Schatten class properties of the associated resolvent operator.Comment: 8 pages

On the eigenvalues of spectral gaps of matrix-valued Schrödinger operators

This paper presents a method for calculating eigenvalues lying in the gaps of the essential spectrum of matrix-valued Schrödinger operators. The technique of dissipative perturbation allows eigenvalues of interest to move up the real axis in order to achieve approximations free from spectral pollution. Some results of the behaviour of the corresponding eigenvalues are obtained. The effectiveness of this procedure is illustrated by several numerical examples

Dogmatism, Learning and Scientific Pratices

In the traditional debate on the dichotomy between dogmatism and criticism in scientific practice (the Popper-Kuhn debate), dogmatism is considered a psychological or ethical attitude of the individual scientist. In this paper, I propose a new interpretation of scientific dogmatism by means of a reconstruction of the pragmatist and Wittgensteinian heritage of Kuhn’s concept of dogmatism. My thesis is that such a revised concept accounts for both the stability of scientific knowledge (against scepticism and ceaseless scientific revolutions), and the importance of doubt and criticism for scientific progress. This is possible only if we consider dogmatism from a social perspective that focuses on scientific communities as the main actors in the history of science. From this point of view, dogmatism is not the unjustified acceptance of particular beliefs, but the blind adherence to the “formal” normative structure of the paradigm. I argue that we have grounded this normative structure in the training process that physics students experience in their formative years, and that Kuhn describes in a similar way to Wittgenstein’s analysis of general linguistic training. Finally, dogmatism does not refer to a system of beliefs, but to a system of norms; not to the specific content of knowledge but to the way that scientific knowledge is authenticated, organised, and transmitted by scientific communities. The institutional structure of science, that is to say its social organisation trough training, textbooks, scientific communities and so on, is a precondition for the organisation of meaningful scientific discourse (i.e. for the production and organisation of empirically verifiable or falsifiable statements). That is the nature of the paradigm: it creates and constrains the possibilities of scientific practice. In normal circumstances, dogmatism and certainty are concerned with such pragmatic a priori, while criticism and doubt are concerned with the empirical statements articulated through it

Computing the Sound of the Sea in a Seashell

The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is $\mathcal C^2$. The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.Comment: 25 pages, 12 figure

The functional model for maximal dissipative operators (translation form): An approach in the spirit of operator knots

In this article we develop a functional model for a general maximal dissipative operator. We construct the selfadjoint dilation of such operators. Unlike previous functional models, our model is given explicitly in terms of parameters of the original operator, making it more useful in concrete applications. For our construction we introduce an abstract framework for working with a maximal dissipative operator and its anti-dissipative adjoint and make use of the ˇStraus characteristic function in our setting. Explicit formulae are given for the selfadjoint dilation, its resolvent, a core and the completely non-selfadjoint subspace; minimality of the dilation is shown. The abstract theory is illustrated by the example of a Schrödinger operator on a half-line with dissipative potential, and boundary condition and connections to existing theory are discussed

Essential spectrum for dissipative Maxwell equations in domains with cylindrical ends

We consider the Maxwell equations with anisotropic coefficients and non-trivial conductivity in a domain with finitely many cylindrical ends. We assume that the conductivity vanishes at infinity and that the permittivity and permeability tensors converge to non-constant matrices at infinity, which coincide with a positive real multiple of the identity matrix in each of the cylindrical ends. We establish that the essential spectrum of Maxwell system can be decomposed as the union of the essential spectrum of a bounded multiplication operator acting on gradient fields, and the union of the essential spectra of the Maxwell systems obtained by freezing the coefficients to their different limiting values along the several different cylindrical ends of the domain