Periodic potentials and supersymmetry

Khare, Avinash; Sukhatme, U.

doi:10.1088/0305-4470/37/43/002

Cited by 32 publications

(39 citation statements)

References 25 publications

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“…However, as has been noticed recently [10], if we consider PT-symmetric complex potentials, then the singularity is not on the real axis. Besides, as we have stressed previously [4,11], in the case of doubly periodic potentials composed of Jacobi elliptic functions, both V (x) and V P T (x)…”

Section: Introductionmentioning

confidence: 65%

PT-invariant periodic potentials with a finite number of band gaps

Khare

Sukhatme

2005

Journal of Mathematical Physics

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Abstract:We obtain the band edge eigenstates and the mid-band states for the complex, PT-invariant generalized associated Lamé potentials, where y ≡ ix+β, and there are four parameters a, b, f, g. This work is a substantial generalization of previous work with the associated Lamé potentials V (x) = a(a + 1)msn 2 (x, m) + b(b + 1)msn 2 (x + K(m), m) and their corresponding PT-invariant counterparts V P T (x) = −V (ix + β), both of which involving just two parameters a, b. We show that for many integer values of a, b, f, g, the PT-invariant potentials V P T (x) are periodic problems with a finite number of band gaps. Further, using supersymmetry, we construct several additional, new, complex, PTinvariant, periodic potentials with a finite number of band gaps. We also point out the intimate connection between the above generalized associated Lamé potential problem and Heun's differential equation.

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Section: Introductionmentioning

confidence: 65%

PT-invariant periodic potentials with a finite number of band gaps

Khare

Sukhatme

2005

Journal of Mathematical Physics

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“…It may be noted here that in I we used the notation [a(a + There is one important point which permits us to construct many supersymmetric partner potentials corresponding to a given potential [5]. Although this point was previously made in I, it is worth restating here.…”

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confidence: 99%

Complex periodic potentials with a finite number of band gaps

Khare

Sukhatme

2006

Journal of Mathematical Physics

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We obtain several new results for the complex generalized associated Lamé potential, sn(y, m) is a Jacobi elliptic function with modulus parameter m, and there are four real parameters a, b, f, g. First, we derive two new duality relations which, when coupled with a previously obtained duality relation, permit us to relate the band edge eigenstates of the 24 potentials obtained by permutations of the four parameters a, b, f, g. Second, we pose and answer the question: how many independent potentials are there with a finite number "a" of band gaps when a, b, f, g are integers?For these potentials, we clarify the nature of the band edge eigenfunctions. We also obtain several analytic results when at least one of the four parameters is a half-integer. As a by-product, we also obtain new solutions of Heun's differential equation.1

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“…Internal rotation, molecular torsion [5][6][7][8][9][10] Uniaxial paramagnets or single-molecule magnets [11][12][13] Molecular alignment/orientation [14,15] Quasi-exactly solvable double well potential [16][17][18] Molecules in combined fields [3] Quantum field theory [19][20][21] Band structure of condensed matter systems [16,17,22] PT-symmetry [23] Nonlinear dynamics [24,25] Nonlinear coherent structures [26,27] Solitons [28,29] Quantum theory of instantons [29][30][31] Josephson Junction Rhombus [32] Model of a proton in a hydrogen bond [17,21,33] Optical lattice [34] each of them associated with one of the four irreducible representations of the C 2v point group. For each of the irreducible representations, the Hamiltonian of the planar pendulum is found to be an infinite tridiagonal matrix containing a finite-dimensional block characterised by a particular condition imposed on the pendulum's parameters and expressed in terms of an integer, termed the topological index.…”

Section: Problems and Applications Reference Problems And Applicationmentioning

confidence: 99%

Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart

et al. 2017

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Abstract.We have subjected the planar pendulum eigenproblem to a symmetry analysis with the goal of explaining the relationship between its conditional quasi-exact solvability (C-QES) and the topology of its eigenenergy surfaces, established in our earlier work [Front. Phys. Chem. Chem. Phys. 2, 1 (2014)]. The present analysis revealed that this relationship can be traced to the structure of the tridiagonal matrices representing the symmetry-adapted pendular Hamiltonian, as well as enabled us to identify many more -40 in total to be exact -analytic solutions. Furthermore, an analogous analysis of the hyperbolic counterpart of the planar pendulum, the Razavy problem, which was shown to be also C-QES [Am. J. Phys. 48, 285 (1980)], confirmed that it is anti-isospectral with the pendular eigenproblem. Of key importance for both eigenproblems proved to be the topological index κ, as it determines the loci of the intersections (genuine and avoided) of the eigenenergy surfaces spanned by the dimensionless interaction parameters η and ζ. It also encapsulates the conditions under which analytic solutions to the two eigenproblems obtain and provides the number of analytic solutions. At a given κ, the anti-isospectrality occurs for single states only (i.e., not for doublets), like C-QES holds solely for integer values of κ, and only occurs for the lowest eigenvalues of the pendular and Razavy Hamiltonians, with the order of the eigenvalues reversed for the latter. For all other states, the pendular and Razavy spectra become in fact qualitatively different, as higher pendular states appear as doublets whereas all higher Razavy states are singlets.

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Periodic potentials and supersymmetry

Cited by 32 publications

References 25 publications

PT-invariant periodic potentials with a finite number of band gaps

PT-invariant periodic potentials with a finite number of band gaps

Complex periodic potentials with a finite number of band gaps

Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart

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