44 research outputs found

    New Solvable and Quasi Exactly Solvable Periodic Potentials

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    Using the formalism of supersymmetric quantum mechanics, we obtain a large number of new analytically solvable one-dimensional periodic potentials and study their properties. More specifically, the supersymmetric partners of the Lame potentials ma(a+1)sn^2(x,m) are computed for integer values a=1,2,3,.... For all cases (except a=1), we show that the partner potential is distinctly different from the original Lame potential, even though they both have the same energy band structure. We also derive and discuss the energy band edges of the associated Lame potentials pm sn^2(x,m)+qm cn^2(x,m)/ dn^2(x,m), which constitute a much richer class of periodic problems. Computation of their supersymmetric partners yields many additional new solvable and quasi exactly solvable periodic potentials.Comment: 24 pages and 10 figure

    Potentials with Two Shifted Sets of Equally Spaced Eigenvalues and Their Calogero Spectrum

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    Motivated by the concept of shape invariance in supersymmetric quantum mechanics, we obtain potentials whose spectrum consists of two shifted sets of equally spaced energy levels. These potentials are similar to the Calogero-Sutherland model except the singular term αx−2\alpha x^{-2} always falls in the transition region −1/4<α<3/4-1/4 < \alpha < 3/4 and there is a delta-function singularity at x=0.Comment: Latex, 12 pages, Figures available from Authors, To appear in Physics Letters A. Please send requests for figures to [email protected] or [email protected]

    Some Exact Results for Mid-Band and Zero Band-Gap States of Associated Lame Potentials

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    Applying certain known theorems about one-dimensional periodic potentials, we show that the energy spectrum of the associated Lam\'{e} potentials a(a+1)m sn2(x,m)+b(b+1)m cn2(x,m)/dn2(x,m)a(a+1)m~{\rm sn}^2(x,m)+b(b+1)m~{\rm cn}^2(x,m)/{\rm dn}^2(x,m) consists of a finite number of bound bands followed by a continuum band when both aa and bb take integer values. Further, if aa and bb are unequal integers, we show that there must exist some zero band-gap states, i.e. doubly degenerate states with the same number of nodes. More generally, in case aa and bb are not integers, but either a+ba + b or a−ba - b is an integer (a≠ba \ne b), we again show that several of the band-gaps vanish due to degeneracy of states with the same number of nodes. Finally, when either aa or bb is an integer and the other takes a half-integral value, we obtain exact analytic solutions for several mid-band states.Comment: 18 pages, 2 figure

    Methods for Generating Quasi-Exactly Solvable Potentials

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    We describe three different methods for generating quasi-exactly solvable potentials, for which a finite number of eigenstates are analytically known. The three methods are respectively based on (i) a polynomial ansatz for wave functions; (ii) point canonical transformations; (iii) supersymmetric quantum mechanics. The methods are rather general and give considerably richer results than those available in the current literature.Comment: 12 pages, LaTe

    Non-Central Potentials and Spherical Harmonics Using Supersymmetry and Shape Invariance

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    It is shown that the operator methods of supersymmetric quantum mechanics and the concept of shape invariance can profitably be used to derive properties of spherical harmonics in a simple way. The same operator techniques can also be applied to several problems with non-central vector and scalar potentials. As examples, we analyze the bound state spectra of an electron in a Coulomb plus an Aharonov-Bohm field and/or in the magnetic field of a Dirac monopole.Comment: Latex, 12 pages. To appear in American Journal of Physic