158 research outputs found

    Sextic anharmonic oscillators and orthogonal polynomials

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    Under certain constraints on the parameters a, b and c, it is known that Schroedinger's equation -y"(x)+(ax^6+bx^4+cx^2)y(x) = E y(x), a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this article we show that the exact wave function y is the generating function for a set of orthogonal polynomials P_n^{(t)}(x) in the energy variable E. Some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi-exact solvability. We also prove that this set of orthogonal polynomials can be reduced,by means of a simple scaling transformation, to a remarkable class of orthogonal polynomials, P_n(E)=P_n^{(0)}(E) recently discovered by Bender and Dunne.Comment: 11 page

    Towards Sustainable and Intelligent Machining:Energy Footprint and Tool Condition Monitoring for Media-Assisted Processes

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    Reducing energy consumption is a necessity towards achieving the goal of net-zero manufacturing. In this paper, the overall energy footprint of machining Ti-6Al-4V using various cooling/lubrication methods is investigated taking the embodied energy of cutting tools and cutting fluids into account. Previous studies concentrated on reducing the energy consumption associated with the machine tool and cutting fluids. However, the investigations in this study show the significance of the embodied energy of cutting tool. New cooling/lubrication methods such as WS2-oil suspension can reduce the energy footprint of machining through extending tool life. Cutting tools are commonly replaced early before reaching their end of useful life to prevent damage to the workpiece, effectively wasting a portion of the embodied energy in cutting tools. A deep learning method is trained and validated to identify when a tool change is required based on sensor signals from a wireless sensory toolholder. The results indicated that the network is capable of classifying over 90% of the tools correctly. This enables capitalising on the entirety of a tool’s useful life before replacing the tool and thus reducing the overall energy footprint of machining processes

    Towards Sustainable and Intelligent Machining:Energy Footprint and Tool Condition Monitoring for Media-Assisted Processes

    Get PDF
    Reducing energy consumption is a necessity towards achieving the goal of net-zero manufacturing. In this paper, the overall energy footprint of machining Ti-6Al-4V using various cooling/lubrication methods is investigated taking the embodied energy of cutting tools and cutting fluids into account. Previous studies concentrated on reducing the energy consumption associated with the machine tool and cutting fluids. However, the investigations in this study show the significance of the embodied energy of cutting tool. New cooling/lubrication methods such as WS2-oil suspension can reduce the energy footprint of machining through extending tool life. Cutting tools are commonly replaced early before reaching their end of useful life to prevent damage to the workpiece, effectively wasting a portion of the embodied energy in cutting tools. A deep learning method is trained and validated to identify when a tool change is required based on sensor signals from a wireless sensory toolholder. The results indicated that the network is capable of classifying over 90% of the tools correctly. This enables capitalising on the entirety of a tool’s useful life before replacing the tool and thus reducing the overall energy footprint of machining processes

    Study of the Generalized Quantum Isotonic Nonlinear Oscillator Potential

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    We study the generalized quantum isotonic oscillator Hamiltonian given by H=−d2/dr2+l(l+1)/r2+w2r2+2g(r2−a2)/(r2+a2)2, g>0. Two approaches are explored. A method for finding the quasipolynomial solutions is presented, and explicit expressions for these polynomials are given, along with the conditions on the potential parameters. By using the asymptotic iteration method, we show how the eigenvalues of this Hamiltonian for arbitrary values of the parameters g, w, and a may be found to high accuracy

    Asymptotic iteration method for eigenvalue problems

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    An asymptotic interation method for solving second-order homogeneous linear differential equations of the form y'' = lambda(x) y' + s(x) y is introduced, where lambda(x) \neq 0 and s(x) are C-infinity functions. Applications to Schroedinger type problems, including some with highly singular potentials, are presented.Comment: 14 page

    Coulomb plus power-law potentials in quantum mechanics

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    We study the discrete spectrum of the Hamiltonian H = -Delta + V(r) for the Coulomb plus power-law potential V(r)=-1/r+ beta sgn(q)r^q, where beta > 0, q > -2 and q \ne 0. We show by envelope theory that the discrete eigenvalues E_{n\ell} of H may be approximated by the semiclassical expression E_{n\ell}(q) \approx min_{r>0}\{1/r^2-1/(mu r)+ sgn(q) beta(nu r)^q}. Values of mu and nu are prescribed which yield upper and lower bounds. Accurate upper bounds are also obtained by use of a trial function of the form, psi(r)= r^{\ell+1}e^{-(xr)^{q}}. We give detailed results for V(r) = -1/r + beta r^q, q = 0.5, 1, 2 for n=1, \ell=0,1,2, along with comparison eigenvalues found by direct numerical methods.Comment: 11 pages, 3 figure

    Energies and wave functions for a soft-core Coulomb potential

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    For the family of model soft Coulomb potentials represented by V(r) = -\frac{Z}{(r^q+\beta^q)^{\frac{1}{q}}}, with the parameters Z>0, \beta>0, q \ge 1, it is shown analytically that the potentials and eigenvalues, E_{\nu\ell}, are monotonic in each parameter. The potential envelope method is applied to obtain approximate analytic estimates in terms of the known exact spectra for pure power potentials. For the case q =1, the Asymptotic Iteration Method is used to find exact analytic results for the eigenvalues E_{\nu\ell} and corresponding wave functions, expressed in terms of Z and \beta. A proof is presented establishing the general concavity of the scaled electron density near the nucleus resulting from the truncated potentials for all q. Based on an analysis of extensive numerical calculations, it is conjectured that the crossing between the pair of states [(\nu,\ell),(\nu',\ell')], is given by the condition \nu'\geq (\nu+1) and \ell' \geq (\ell+3). The significance of these results for the interaction of an intense laser field with an atom is pointed out. Differences in the observed level-crossing effects between the soft potentials and the hydrogen atom confined inside an impenetrable sphere are discussed.Comment: 13 pages, 5 figures, title change, minor revision

    Solutions for certain classes of Riccati differential equation

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    We derive some analytic closed-form solutions for a class of Riccati equation y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are C^{\infty}-functions. We show that if \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}= \lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1} and s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1}, n=1,2,..., then The Riccati equation has a solution given by y(x)=\mp s_{n-1}(x)/\lambda_{n-1}(x). Extension to the generalized Riccati equation y'(x)+P(x)y(x)+Q(x)y^2(x)=R(x) is also investigated.Comment: 10 page

    Criterion for polynomial solutions to a class of linear differential equation of second order

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    We consider the differential equations y''=\lambda_0(x)y'+s_0(x)y, where \lambda_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential equation, has a polynomial solution of degree n >0, then \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}= \lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1}\hbox{and}\quad s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1},\quad n=1,2,.... Conversely (ii) if \lambda_n\lambda_{n-1}\ne 0 and \delta_n=0, then the differential equation has a polynomial solution of degree at most n. We show that the classical differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first and second kind), Gegenbauer, and the Hypergeometric type, etc, obey this criterion. Further, we find the polynomial solutions for the generalized Hermite, Laguerre, Legendre and Chebyshev differential equations.Comment: 12 page
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