Schrödinger equation with potential function vanishing exponentially fast

Explicitly oscillating solutions of differential equation and its eigenpairs are obtained by calculating complex residues. Eigenpairs, spectral function and eigenfunction expansions are also reported for this specific Pöschl–Teller differential equation.


Introduction
In this paper, we consider the following second-order differential equation: with specific potential function q(x) = −20 sech 2 x. Theory related to (1) is elaborately given in [1]. One may say that the only case to solve (1) explicitly for all λ is the case q(x) = 0 where the solutions are oscillating. In reality, this is not correct. There are indeed some examples which are not reported yet. One such example is y (x) + λ + 20 sech 2 x y(x) = 0, which is a special case of y (x) + λ + n(n + 1) sech 2 x y(x) = 0 with n = 4. This example is slightly different from what Titchmarsh does in the following example where its solutions involving Legendre functions For similar examples and results, see [1][2][3][4][5][6][7][8].
It is worth mentioning here how our interest in this title arises. There is a relationship between the Schrödinger equation and Sturm-Liouville differential equation (1). The Schrödinger equation arises from several partial differential equation appearing in physics by separation of variables [9]. For instance, for a single quantum-mechanical particle of mass m moving in one space dimension in a potential V(x), the timedependent Schrödinger equation is Looking for separable solutions where E is energy eigenstates. Then one finds that After normalization, one obtains Equation (1) in the subject line above where the potential function q = (2m/ 2 )V(x), the eigenvalue parameter λ = (2m/ 2 )E and y = ϕ. Therefore, if one considers x → ∞, then Hence, one may see that why we called title in the subject line above. It is also important to note that Equation (2) describes the diatomic molecular vibration and such potential also arises in the solutions of the Korteweg-de Vries equation [10,11].
In general, it is not easy to obtain the solutions of Equation (1) as we obtain in our case by employing residues. Conventionally, solutions to Equation (1) are written either in terms of hypergeometric functions or as series. The purpose of this article is to provide analytical solutions of Equation (2) by using local complex residues.

Preliminaries
We now give some preliminaries. To get the expansion of an arbitrary function f (x) in terms of eigenfunctions one actually needs to know the following definitions and lemmas taken from [1,2]. If θ(x, λ) and φ(x, λ) are the solutions of (1), with α is real, satisfying Then Wronskian W x (θ , φ) = sin 2 α + cos 2 α = 1. It is well known that the general solution of (1) is The definition of the non-decreasing spectrum function is given by where , .

Main results
We are now aiming to deal with the solution of (2) which is given by with s 2 = −λ where C is just including the point z = x and excluding the other zeros of sinh z − sinh x. Proof: Applying two times partial integration to (5) successively, one obtains By using s 2 = −λ, Comparing (6) and (7), we see that

Proof:
The proof is the same as Theorem 3.1. Hence, it is excluded.

Eigenpairs obtained by calculating the relevant residues
As a result of very laborious calculations, one obtains that the residue at z = x for (5)  Hence, by using s 2 = −λ, one solution is Similarly, by examining the residue of (8), one obtains second solution as where y 1 (x) and y 2 (x) are given by (5) and (8) respectively, then After all, and C sin(s−1)z cos(z) + sin(z) cos(s−1)z cos 5 z dz, partial integral implies that keep continuing this argument, That is, By combining I 1 and I 2 , one gets the expected result.

Eigenfunction expansions
If θ(x, λ) and φ(x, λ) are the solutions of (2) and satisfying (3). Then one finds (10) One needs to find spectral function k(λ). To get desired result, we have to find the asymptotics of (10) as x → ∞. If Im( √ λ) > 0, then the asymptotics are where where After arranging that the linear combination of (11) and (12), then the terms e −ix √ λ cancel out.

Conclusion
The explicit solutions of the Pöschl-Teller differential equation (2) and its eigenpairs are obtained by calculating complex residues. Eigenfunction expansions for arbitrary function f (x) satisfying suitable conditions are also obtained. We concluded that differential equation (2) has oscillating solutions which are not reported in the literature.

Disclosure statement
No potential conflict of interest was reported by the author.