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Exact Methods for ODEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
In supersymmetric quantum mechanics, a superpotential (W) defines the supersymmetric partner potentials (V±) via V±=W2(x)±W′(x). The corresponding Hamiltonians (H±) factor as H±=A∓A± where A±=∓ddx+W(x). If Ψn± is an eigenfunction of H± (that is, H±Ψn±=En±Ψn± ), then we have the ladder operations Ψn+∝A−Ψn+1− and Ψn+1−∝A+Ψn+. For example, for the simple harmonic oscillator W(x)=x−2bω and V−=14ω2x−2bω2−12ω. Dabrowska et al.[306] contains a list of superpotentials and associated information.
An error estimate for bilateral contact problem with nonmonotone friction between two electroelastic bodies
Published in Applicable Analysis, 2023
In this work, we apply of Riesz's representation theorem and the Gauss elimination technique and then we decouple the mechanical field from the electric field analytically. The resulting problem is a perturbed mixed formulation arising in elastic contact problems. The existence is established and the uniqueness proved under an additional condition, as in [33]. To decouple the linear elasticity from friction, the non-convex superpotential theory, see e.g. [11,21,36,37], is used and then we obtain a mixed formulation of the problem. Furthermore, to approximate the problem we employ two independent mesh sizes with regard to each domain. We then consider two cases, the first is when the mesh sizes are equal which leads to the conformal approximation. In the second one, we use two different mesh sizes and thus a to non-conformal approximation. The later case is more complex but more suitable for numerical handling of the problem using parallel computing. Once the problem is approximated, we obtain error estimates for both approximations. To this end, we use some properties of the projection operator [31,32] and of a superpotential (see [11]and Khenous et al. [37]). Thena matrix formulation is developed.
Coupled supersymmetry and ladder structures beyond the harmonic oscillator
Published in Molecular Physics, 2018
Cameron L. Williams, Nikhil N. Pandya, Bernhard G. Bodmann, Donald J. Kouri
Definition 1 includes the QMHO by letting , , . There exists an infinite family of examples solving the coupled SUSY equations. A straightforward calculation shows that, for , the operators taken with their adjoints on also define a coupled SUSY when restricted to an appropriate subspace where and . This family of examples is closely related to standard SUSY with the anharmonic superpotential where the charge operator has been multiplied on the left by . While a change of variable relates these operators to the traditional QMHO ladder operators, their respective adjoints have a much different form, creating an altogether new system.