Pt Symmetry and Supersymmetry 1. Supersymmetry for Pedestrians, or Bosons and Fermions in Witten's Picture
Pseudo-Hermitian (so called PT symmetric) Hamiltonians are featured within a re-formulated Witten's supersymmetric quantum mechanics. An unusual form of the supersymmetric partnership between the spiked harmonic oscillators is described. In the review [1] of the Witten's (or so called supersymmetric) quantum mechanics one finds the specific Fock vacuum q|0 = exp(−q 2 /2)/ √ π 0 , q ∈ (−∞, ∞). (1) In the upper line we recognize the ground state of the one-dimensional harmonic oscillator H (HO) = p 2 + q 2 and see that " bosons " may be created/annihilated by the action of the respective operators −∂ r + q and ∂ r + q (to be denoted as B (−1/2) and A (−1/2) here). The solvable harmonic-oscillator character of this elementary model enables us to define the three two-by-two matrices F † = 0 0 1 0 , F = 0 1 0 0 , N F = 0 0 0 1 (2) which create and annihilate a " fermion " and determine the fermionic number, respectively. Formally, this enables us to introduce the two partner Hamiltonians H (L) = H (HO) − 1, H (R) = H (HO) + 1 and identify the underlying symmetry with the superalgebra sl(1/1) generated by the following three operator matrices
Supersymmetry in the half oscillator: Revisited
Following a recent study by Das and Pernice [Nucl. Phys. B561, (1999) 357], we have carefully analyzed the half-harmonic oscillator. In contrast to their observations, our analysis reveals that the spectrum does not allow for a zero energy ground state and hence the supersymmetry is broken when the domain is restricted to the positive half of the real axis.
Multilayer high density flex technology
The industrial trend of shrinking microelectronic devices while increasing the density of interconnections places great demands on the substrates upon which these devices are packaged. Multilayer flex circuits will provide the interconnection densities needed to meet these demands in a wide range of packaging and interconnection technologies. This paper discusses the development of ultra-high density flexible circuits. Processes and materials to fabricate fine line pitches from 12.5 /spl mu/m to 40 /spl mu/m are examined. Various microvia structures, through hole and blind via, are demonstrated. A test vehicle was designed and built for BGA packages to illustrate the density capability. A four-layer structure is demonstrated by using Z-connection to connect two layer pairs. Electrical, mechanical, and reliability test results are presented to show the selection of flex substrate material, Z-connection methodology and fabrication processes. A cost comparison of high-density flex to alternative substrates is discussed to identify potential.