Solvable models of an open well and a bottomless barrier: one-dimensional exponential potentials

Abstract

We present one dimensional potentials V (x) = V 0 [e 2|x|/a − 1] as solvable models of a well (V 0 > 0) and a barrier (V 0 < 0). Apart from being new addition to solvable models, these models are instructive for finding bound and scattering states from the analytic solutions of Schrödinger equation. The exact analytic (semi-classical and quantal) forms for bound states of the well and reflection/transmission (R/T ) coefficients for the barrier have been derived. Interestingly, the crossover energy E c where R(E c ) = 1/2 = T (E c ) may occur below/above or at the barrier-top.A connection between poles of these coefficients and bound state eigenvalues of the well has also been demonstrated. * Electronic address: 1:zahmed@barc.gov.in, 2

Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kiν(z) with Respect to Order

“…A generalization of the respective approaches for the positive sign in the exponential is straightforward and can be considered a textbook problem, cf. [2,22]. Below we will briefly recall the derivation chain for consistency.…”

“…that provides only a slightly better extimation comparing to equation ( 7), as can be seen from Figure 3. The use of a similar function was suggested in [2]. 9)-( 10) for zeros ν n of K iν (1).…”

Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of $K_{{\rm i}ν}(z)$ with Respect to Order

“…A generalization of the respective approaches for the positive sign in the exponential is straightforward and can be considered a textbook problem, cf. [21,22]. Below we will briefly recall the derivation chain for consistency.…”

On beautiful analytic structure of the S-matrix

“…Any difference between the respective poles gets blurred only at the level of the S-matrix when the ratio as the basis of linearly independent solutions of equation (4) in exponentially attractive and repulsive cases, in spite that each of them collapses into linearly dependent solutions for any  r Î i (see equation (15)). Those choices goes back to the classical contributions of Ma [5][6][7] and stretch, for instance, to recent treatments of (i) one-dimensional exponential potentials V(x) on Î -¥ ¥ ( ) x , [26] and (ii) scattering and bound states in scalar and vector exponential potentials in the Klein-Gordon equation [ . The latter basis never degenerate into linearly dependent solutions and is standard choice when treating electromagnetic scattering from dielectric objects [28,29].…”