Hidden nonlinear supersymmetry of finite-gap Lamé equation

Abstract

A bosonized nonlinear (polynomial) supersymmetry is revealed as a hidden symmetry of the finite-gap Lamé equation. This gives a natural explanation for peculiar properties of the periodic quantum system underlying diverse models and mechanisms in field theory, nonlinear wave physics, cosmology and condensed matter physics. PACS numbers: 11.30.Pb; 11.30.Na; 03.65.Fd Supersymmetry [1], as a fundamental symmetry providing a natural mechanism for unification of gravity with electromagnetic, strong and weak interactions, still waits for experimental confirmation. On the other hand, in nuclear physics supersymmetry was predicted theoretically [2] and has been confirmed experimentally [3] as a dynamic symmetry linking properties of some bosonic and fermionic nuclei. It would be interesting to look for some physical systems whose special properties could be explained by a hidden ordinary (not a dynamic) supersymmetry.In the present Letter we show that the quantum system described by the finite-gap Lamé equation possesses a hidden supersymmetry. A very unusual nature of the revealed supersymmetry is that it manifests as a nonlinear symmetry of a bosonic system without fermion (spin) degrees of freedom. This means that we find here a kind of a bosonized supersymmetry giving a natural explanation for peculiar properties of the periodic quantum problem underlying many physical systems.The Lamé equation first arose in solution of the Laplace equation by separation of variables in ellipsoidal coordinates [4], and one of its early applications was in the quantum Euler top problem [5]. It plays nowadays a prominent role in physics appearing in such diverse theories as crystals models in solid state physics [6,7], exactly and quasi-exactly solvable quantum systems [8,9], integrable systems and solitons [10,11] [21], and preheating after inflation modern theories [22]. Most often, the Lamé equation appears in physics literature in the Jacobian form of a one-dimensional Schrodinger equation with a doubly periodic potential,where sn(x, k) ≡ sn x is the Jacobi elliptic odd function with modulus k (0 < k < 1), and real and imaginary periods 4K and 2iK ′ , K = K(k) is a complete elliptic integral of the first kind, and 4,23]. A remarkable property of this equation is that at integer values of the parameter j = n, its energy spectrum has exactly n gaps, which separate the n+1 allowed energy bands. The 2n + 1 eigenfunctions associated to the boundaries E i (n), i = 0, 1, . . . , 2n, of the allowed energy bands, ∞] are given by polynomials ('Lamé polynomials') of degree n in the elliptic functions sn x, cn x and dn x. These polynomials have real periods 4K or 2K, and the boundary energy levels E i (n) are non-degenerated. The states in the interior of allowed zones are described by the quasi-periodic Bloch-Floquet wave functions (which can be expressed in terms of theta functions [4]) of quasi-momentum κ(E),Every such interior energy level is doubly degenerated. For any non-integer value of the parameter j, Eq. (1) has an infinite num...

Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials

“…48 Potentials with elliptic functions can also be discussed from this point of view. 49 Potentials with elliptic functions appear in Ref. 9.…”

Exotic supersymmetry of the kink-antikink crystal, and the infinite period limit

“…(5.19), (5.21), and (6.5), PðS 1 ; Þ is the sixth-order spectral polynomial of the BdG system, PðS 1 ;Þ ¼ ðS 2 1 À"ðÞÞðS 2 1 À"ðÞÀk 02 ÞðS 2 1 À"ðÞÀ1Þ; (7.4) whose six roots correspond to the energy levels (7.1). Superalgebra (7.3) has a structure similar to that of a hidden, bosonized supersymmetry [47] of the unextended Lamé system (2.1), which was revealed in [38]. There, the role of the grading operator is played by a reflection operator R, the matrix integrals L a are substituted by the Lax operator ÀiP ðxÞ, see Eq.…”

Soliton defects in one-gap periodic system and exotic supersymmetry

“…If we restrict the parameters κ 1,2 by the condition 0 < κ 1 < κ 2 < 1, the corresponding Wronskian W(x) = W (φ 1 , φ 2 ) has no zeros. This produces a system with a regular reflectionless potential 18) which has three bound states with energies 1 − κ 2 1 , 1 − κ 2 2 and 0. Sending then one of the two translation parameters, τ 2 or τ 1 , to any of the limits +∞ or −∞, we get a reflectionless system with two bound states of energies 1 − κ 2 1 and 0 when we send |τ 2 | → ∞, or with energies 1 − κ 2 2 and 0 when |τ 1 | → ∞.…”

“…The Lamé system's integral P 0,0 satisfies the Burchnall-Chaundy relation 26) which lies in the basis of the hidden bosonized nonlinear supersymmetry of the one-gap Lamé system [18]. The zeros of the third order polynomials in H 0,0 correspond to the energies of the edges of the allowed bands of (2.1).…”

“…The isospectral Hamiltonians H 0,0 (x) and H 0,0 (x + β − ) are connected by the first order differential operator 18) and by its Hermitian conjugate operator, where…”