Abstract:We present the general form of potentials with two given energy levels E 1 , E 2 and find corresponding wave functions. These entities are expressed in terms of one function ξ(x) and one parameter ∆E = E 2 -E 1 . We show how the quantum numbers of both levels depend on properties of the function ξ(x).Our approach does not need resorting to the technique of supersymmetric (SUSY) quantum mechanics but generates the expression for the superpotential automatically.
“…Choosing generating functions φ(x) with one zero we obtained QES potentials with explicitly know ground and first excited states. Note that basic equation derived by Dolya and Zaslavskii in [19] without resorting to SUSY quantum mechanics are the same as was earlier obtained in [21] using SUSY method (in [19] φ(x) is denoted as ξ(x)). A new result obtained by Dolya and Zaslavskii is that they have shown how one can obtain not only the ground and first excited states but any pair of states using generating function φ(x) and their derivative φ ′ (x) with zeros and poles.…”
Section: Solutions For Superpotentials and Construction Of Nonsingulamentioning
confidence: 80%
“…Within the frames of this method we have obtained QES potentials for which we have found in the explicit form the energy levels and wave functions of the ground and first excited states. One should mention here also paper [18] where the general expression for the QES potentials with two known eigenstates was obtained without resorting to the SUSY quantum mechanics (see also a very recent paper by Dolya and Zaslavskii [19]). Although this method is direct and simpler than the SUSY approach the latter still has some advantages.…”
Using supersymmetric quantum mechanics we construct the quasiexactly solvable (QES) potentials with arbitrary two known eigenstates. The QES potential and the wave functions of the two energy levels are expressed by some generating function the properties of which determine the state numbers of these levels. Choosing different generating functions we present a few explicit examples of the QES potentials.
“…Choosing generating functions φ(x) with one zero we obtained QES potentials with explicitly know ground and first excited states. Note that basic equation derived by Dolya and Zaslavskii in [19] without resorting to SUSY quantum mechanics are the same as was earlier obtained in [21] using SUSY method (in [19] φ(x) is denoted as ξ(x)). A new result obtained by Dolya and Zaslavskii is that they have shown how one can obtain not only the ground and first excited states but any pair of states using generating function φ(x) and their derivative φ ′ (x) with zeros and poles.…”
Section: Solutions For Superpotentials and Construction Of Nonsingulamentioning
confidence: 80%
“…Within the frames of this method we have obtained QES potentials for which we have found in the explicit form the energy levels and wave functions of the ground and first excited states. One should mention here also paper [18] where the general expression for the QES potentials with two known eigenstates was obtained without resorting to the SUSY quantum mechanics (see also a very recent paper by Dolya and Zaslavskii [19]). Although this method is direct and simpler than the SUSY approach the latter still has some advantages.…”
Using supersymmetric quantum mechanics we construct the quasiexactly solvable (QES) potentials with arbitrary two known eigenstates. The QES potential and the wave functions of the two energy levels are expressed by some generating function the properties of which determine the state numbers of these levels. Choosing different generating functions we present a few explicit examples of the QES potentials.
“…Moreover, V k = δ k,0 + δ k,−1 + δ k,−2 and v l,k = 0 but for v l,0 = 3δ l,−2 /4 + δ l,2 + δ l,6 /4, v l,−1 = −96δ l,2 + 16δ l,6 and v l,−2 = −2048δ l, 6 . The cases N = 0 and N = 1 do not lead to a solution.…”
Section: The Kuliy-tkachuk Potentialmentioning
confidence: 99%
“…(6). The generalization consists in using the exponential of a h(x)-series, instead of the differential form given in (9), or, equivalently, to multiply the latter by h ′ (x) and properly redefine the expansion coefficients.…”
Section: A More General Approachmentioning
confidence: 99%
“…-Second, this reduction from differential to algebraic is maintained but the number of known solutions (essentially two [5,6] or three [7]) is fixed at the start. Generally speaking this second point of view is not subtended by any Lie algebra except in the case of the two solutions [8].…”
We construct a general algorithm generating the analytic eigenfunctions as well as eigenvalues of one-dimensional stationary Schrödinger Hamiltonians. Both exact and quasi-exact Hamiltonians enter our formalism but we focus on quasi-exact interactions for which no such general approach has been considered before. In particular we concentrate on a generalized sextic oscillator but also on the Lamé and the screened Coulomb potentials.
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