Functional inversion for potentials in quantum mechanics

Let E = F(v) be the ground-state eigenvalue of the Schroedinger Hamiltonian H = -Delta + vf(x), where the potential shape f(x) is symmetric and monotone increasing for x > 0, and the coupling parameter v is positive. If the 'kinetic potential' bar{f}(s) associated with f(x) is defined by the transformation: bar{f}(s) = F'(v), s = F(v)-vF'(v),then f can be reconstructed from F by the sequence: f^{[n+1]} = bar{f} o bar{f}^{[n]^{-1}} o f^{[n]}. Convergence is proved for special classes of potential shape; for other test cases it is demonstrated numerically. The seed potential shape f^{[0]} need not be 'close' to the limit f.Comment: 14 pages, 2 figure

Spectral bounds for the cutoff Coulomb potential

The method of potential envelopes is used to analyse the bound-state spectrum of the Schroedinger Hamiltonian H = -Delta -v/(r+b), where v and b are positive. We established simple formulas yielding upper and lower energy bounds for all the energy eigenvalues.Comment: 11 pages, 2 figure

Relativistic N-boson systems bound by pair potentials V(r_{ij}) = g(r_{ij}^2)

We study the lowest energy E of a relativistic system of N identical bosons bound by pair potentials of the form V(r_{ij}) = g(r_{ij}^2) in three spatial dimensions. In natural units hbar = c = 1 the system has the semirelativistic `spinless-Salpeter' Hamiltonian H = \sum_{i=1}^N \sqrt{m^2 + p_i^2} + \sum_{j>i=1}^N g(|r_i - r_j|^2), where g is monotone increasing and has convexity g'' >= 0. We use `envelope theory' to derive formulas for general lower energy bounds and we use a variational method to find complementary upper bounds valid for all N >= 2. In particular, we determine the energy of the N-body oscillator g(r^2) = c r^2 with error less than 0.15% for all m >= 0, N >= 2, and c > 0.Comment: 15 pages, 4 figure

Isometry theorem for the Segal-Bargmann transform on noncompact symmetric spaces of the complex type

We consider the Segal-Bargmann transform for a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of singularities allows the relevant integral to have a nonsingular extension to large R, even though the function being integrated has singularities.Comment: Final version. To appear in Journal of Functional Analysis. Minor revision

SO(10) and SU(6) Unified Theories on an Elongated Rectangle

Maximally supersymmetric SO(10) and SU(6) unified theories are constructed on the orbifold T^2/(Z_2 x Z'_2), with one length scale R_5 taken much larger than the other, R_6. The effective theory below 1/R_6 is found to be the highly successful SU(5) theory in 5D with natural doublet-triplet splitting, no proton decay from operators of dimension four or five, unified mass relations for heavier generations only, and a precise prediction for gauge coupling unification. A more unified gauge symmetry, and the possibility of Higgs doublets being components of the higher dimensional gauge multiplet, are therefore compatible with a large energy interval where physics is described by SU(5) gauge symmetry in 5D. This leads to the distinctive branching ratios for proton decay from SU(5) gauge boson exchange, p -> l^+ pi^0, l^+ K^0, \bar{nu} pi^+, \bar{nu} K^+ (l = e, mu), for well-motivated locations for matter. Several phenomenological features of the higher unified gauge symmetry are discussed, including the role of an extra U(1) gauge symmetry, which survives compactification, in the generation of neutrino masses.Comment: 21 pages, LaTe

Klein-Gordon lower bound to the semirelativistic ground-state energy

For the class of attractive potentials V(r) <= 0 which vanish at infinity, we prove that the ground-state energy E of the semirelativistic Hamiltonian H = \sqrt{m^2 + p^2} + V(r) is bounded below by the ground-state energy e of the corresponding Klein--Gordon problem (p^2 + m^2)\phi = (V(r) -e)^2\phi. Detailed results are presented for the exponential and Woods--Saxon potentials.Comment: 7 pages, 4 figure

Safety Engineering with COTS components

Safety-critical systems are becoming more widespread, complex and reliant on software. Increasingly they are engineered through Commercial Off The Shelf (COTS) (Commercial Off The Shelf) components to alleviate the spiralling costs and development time, often in the context of complex supply chains. A parallel increased concern for safety has resulted in a variety of safety standards, with a growing consensus that a safety life cycle is needed which is fully integrated with the design and development life cycle, to ensure that safety has appropriate influence on the design decisions as system development progresses. In this article we explore the application of an integrated approach to safety engineering in which assurance drives the engineering process. The paper re- ports on the outcome of a case study on a live industrial project with a view to evaluate: its suitability for application in a real-world safety engineering setting; its benefits and limitations in counteracting some of the difficulties of safety en- gineering with COTS components across supply chains; and, its effectiveness in generating evidence which can contribute directly to the construction of safety cases

Deaf children need language, not (just) speech

Deaf and Hard of Hearing (DHH) children need to master at least one language (spoken or signed) to reach their full potential. Providing access to a natural sign language supports this goal. Despite evidence that natural sign languages are beneficial to DHH children, many researchers and practitioners advise families to focus exclusively on spoken language. We critique the Pediatrics article ‘Early Sign Language Exposure and Cochlear Implants’ (Geers et al., 2017) as an example of research that makes unsupported claims against the inclusion of natural sign languages. We refute claims that (1) there are harmful effects of sign language and (2) that listening and spoken language are necessary for optimal development of deaf children. While practical challenges remain (and are discussed) for providing a sign language-rich environment, research evidence suggests that such challenges are worth tackling in light of natural sign languages providing a host of benefits for DHH children – especially in the prevention and reduction of language deprivation.Accepted manuscrip

Projective vs metric structures

We present a number of conditions which are necessary for an n-dimensional projective structure (M,[nabla]) to include the Levi-Civita connection nabla of some metric on M. We provide an algorithm, which effectively checks if a Levi-Civita connection is in the projective class and, in the positive, which finds this connection and the metric. The article also provides a basic information on invariants of projective structures, including the treatment via Cartan's normal projective connection. In particular we show that there is a number of Fefferman-like conformal structures, defined on a subbundle of the Cartan bundle of the projective structure, which encode the projectively invariant information about (M,[nabla])