[Paper Review] Confining non-analytic exponential potential $V(x)= g^2\exp\,(2|x|)$ and its exact Bessel-function solvability

Ryu Sasaki|arXiv (Cornell University)|Nov 8, 2016

Quantum Mechanics and Non-Hermitian Physics4 references3 citations

TL;DR

This paper presents the exact solvability of the one-dimensional Schrödinger equation with the confining non-analytic exponential potential $V(x) = g^2 \exp(2|x|)$, showing that its bound states are described by modified Bessel functions of the second kind $K_{i\nu}(\rho)$ with pure imaginary order. The energy eigenvalues $E_n = \nu_n^2$ are determined by the pure imaginary zeros of $K_{i\nu}(g)$ and $K'_{i\nu}(g)$, establishing a new class of exactly solvable quantum systems based on Bessel function zeros.

ABSTRACT

In a previous paper we have shown that Schrödinger equation with the non-analytic attractive exponential potential $V(x)= -g^2\exp (-|x|)$ is exactly solvable. It has finitely many discrete eigenstates described by the Bessel function of the first kind $J_ν(z)$ and the eigenvalues are specified by the positive zeros of $J_ν(g)$ and $J'_ν(g)$ as a function of the order $ν$ with fixed $g>0$. Now we show the corresponding results for the {\em confining\/} non-analytic exponential potential $V(x)= g^2\exp (2|x|)$. This has infinitely many discrete eigenstates described by the modified Bessel function of the second kind $K_{iν}(z)$. The eigenvalues are specified by the {\em pure imaginary zeros\/} of $K_{iν}(g)$ and $K'_{iν}(g)$ as a function of the order with fixed $g>0$.

Motivation & Objective

• To establish exact solvability of the confining non-analytic exponential potential $V(x) = g^2 \exp(2|x|)$ in one-dimensional quantum mechanics.
• To derive the complete discrete spectrum of bound states using modified Bessel functions of the second kind with pure imaginary order $i\nu$.
• To identify the energy eigenvalues $E_n = \nu_n^2$ as determined by the zeros of $K_{i\nu}(g)$ and $K'_{i\nu}(g)$ as functions of $\nu$ with fixed $g>0$.
• To derive orthogonality relations and asymptotic distribution laws for the pure imaginary zeros of $K_{i\nu}(g)$ and $K'_{i\nu}(g)$.

Proposed method

• Transform the Schrödinger equation into a Bessel-type differential equation via the auxiliary variable $\rho(x) = g e^{|x|}$, mapping the problem to a radial form.
• Solve the resulting equation using modified Bessel functions $I_{i\nu}(\rho)$ and $K_{i\nu}(\rho)$, with $K_{i\nu}(\rho)$ selected for normalizability due to its decay at infinity.
• Apply parity-invariant boundary conditions at $x=0$ to select even and odd parity eigenfunctions, leading to matching conditions involving $K_{i\nu}(g)$ and $K'_{i\nu}(g)$.
• Determine eigenvalues $E_n = \nu_n^2$ by requiring the eigenfunctions to vanish at the matching point $\rho = g$, i.e., $K_{i\nu}(g) = 0$ or $K'_{i\nu}(g) = 0$.
• Derive orthogonality relations for the eigenfunctions using Wronskian identities and indefinite integration formulas from Watson's treatise on Bessel functions.
• Use the WKB approximation to conjecture the asymptotic distribution of the pure imaginary zeros $\nu_n$ for large $n$, leading to a Bohr-Sommerfeld-type quantization condition.

Experimental results

Research questions

• RQ1 How can the confining exponential potential $V(x) = g^2 \exp(2|x|)$ be exactly solved in one-dimensional quantum mechanics?
• RQ2 What role do modified Bessel functions of the second kind with pure imaginary order $i\nu$ play in describing the bound states of this potential?
• RQ3 How are the energy eigenvalues $E_n = \nu_n^2$ determined by the zeros of $K_{i\nu}(g)$ and $K'_{i\nu}(g)$ as functions of $\nu$ with fixed $g>0$?
• RQ4 What are the orthogonality relations satisfied by the eigenfunctions constructed from $K_{i\nu}(\rho)$?
• RQ5 What is the asymptotic distribution of the pure imaginary zeros $\nu_n$ of $K_{i\nu}(g)$ and $K'_{i\nu}(g)$ for large $n$?

Key findings

• The Schrödinger equation with $V(x) = g^2 \exp(2|x|)$ has infinitely many discrete bound states, with energy eigenvalues $E_n = \nu_n^2$ determined by the zeros of $K_{i\nu}(g)$ and $K'_{i\nu}(g)$ as functions of $\nu$ with fixed $g>0$.
• The eigenfunctions are expressed in terms of the modified Bessel function of the second kind $K_{i\nu}(\rho)$, with $\rho(x) = g e^{|x|}$, and satisfy the required normalizability and parity conditions.
• The zeros $\nu_n$ of $K_{i\nu}(g)$ and $K'_{i\nu}(g)$ are interlaced and simple, with $0 < \lambda_0 < \mu_0 < \lambda_1 < \mu_1 < \cdots$, where $\lambda_n$ and $\mu_n$ are the zeros of $K_{i\nu}(g)$ and $K'_{i\nu}(g)$, respectively.
• Orthogonality relations for the eigenfunctions are established: $\int_x^\infty K_{i\lambda_j}(\rho) K_{i\lambda_k}(\rho) \frac{d\rho}{\rho} = 0$ for $j \neq k$ (even parity), and similarly for odd parity with $\mu_n$.
• A conjecture is proposed for the asymptotic distribution of the $n$-th pure imaginary zero: $\nu_n \,{\rm arccosh}(\nu_n / g) - \sqrt{\nu_n^2 - g^2} = \frac{(n + \frac{1}{2})\pi}{2}$ for $n \gg 1$, consistent with the WKB approximation.
• The eigenfunctions of the $L$-th associated Hamiltonian system satisfy Wronskian-based orthogonality relations involving products of Wronskians of $K_{i\nu_j}(\rho)$, confirming the completeness of the spectrum.