[Paper Review] Adiabatic evolution generated by a one-dimensional Schrödinger operator with decreasing number of eigenvalues

Alexander Fedotov|arXiv (Cornell University)|Sep 29, 2016

Spectral Theory in Mathematical Physics4 references4 citations

TL;DR

This paper studies adiabatic evolution in a one-dimensional Schrödinger operator with a time-dependent potential that causes eigenvalues to approach and disappear into the absolutely continuous spectrum. Using a Sommerfeld-Malyuzhinets-type integral transform and asymptotic analysis, the authors derive precise asymptotic expressions for solutions near the point of eigenvalue disappearance, revealing critical transition dynamics including phase shifts and amplitude decay governed by Airy-type integrals and zeta function contributions.

ABSTRACT

We study a one-dimensional non-stationary Schrödinger equation with a potential slowly depending on time. The corresponding stationary operator depends on time as on a parameter. It has a finite number of negative eigenvalues and absolutely continuous spectrum filling the positive semiaxis. The eigenvalues move with time to the edge of the continuous spectrum and, having reached it, disappear one after another. We study the asymptotic behavior of a solution close at some moment to an eigenfunction of the stationary operator, and, in particular, the phenomena occurring when the corresponding eigenvalue approaches the absolutely continuous spectrum and disappears.

Motivation & Objective

• To analyze the adiabatic evolution of quantum states in a one-dimensional Schrödinger operator with a time-dependent potential that reduces the number of bound states.
• To understand the dynamical behavior of solutions initially close to eigenfunctions when the corresponding eigenvalue approaches and merges into the continuous spectrum.
• To derive rigorous asymptotic expressions for the solution near the critical moment of eigenvalue disappearance, capturing transition phenomena such as phase shifts and amplitude decay.
• To establish a connection between the spectral transition and special functions (e.g., Riemann zeta function, Airy-type integrals) through complex analysis and integral transform methods.

Proposed method

• Construct a generating solution $\Psi$ using an $\varepsilon$-periodic ansatz in momentum space, expressed as a sum over reflected and refracted plane waves.
• Employ an integral transform approach based on the Sommerfeld-Malyuzhinets method to reduce the time-dependent Schrödinger equation to a difference equation in the complex plane with shift parameter $\varepsilon$.
• Solve the functional equation for the amplitude function $R(p)$ via the recurrence $R(p + \varepsilon/2) = \rho(p) R(p - \varepsilon/2)$, with $\rho(p) = (Q(p) - p)/(Q(p) + p)$ and $Q(p) = \sqrt{p^2 - 1}$.
• Analyze the solution $\Psi_n$ via Fourier decomposition, focusing on the behavior inside the time-evolving potential well $0 \leq x \leq 1 - \varepsilon t$.
• Apply complex contour deformation and stationary phase methods to evaluate oscillatory integrals, particularly near the spectral threshold $p=1$.
• Use asymptotic expansions involving the Riemann zeta function $\zeta(s)$ and Airy-type integrals to describe the transition regime as the eigenvalue approaches the continuous spectrum.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of a solution to the time-dependent Schrödinger equation when the corresponding eigenvalue approaches the edge of the absolutely continuous spectrum?
• RQ2 How do the amplitude and phase of the solution evolve during the eigenvalue disappearance process, particularly near the critical time $\tau = \tau_n$?
• RQ3 What role do special functions such as $\zeta(s)$ and $\sqrt{-s}$ play in describing the transition dynamics of the wave function?
• RQ4 How does the solution's structure change from being dominated by a bound state to being influenced by the continuous spectrum?
• RQ5 Can the transition be described using a universal asymptotic form involving Airy functions or related integrals?

Key findings

• The solution $\Psi_n$ exhibits a universal asymptotic form near the eigenvalue disappearance time $\tau = \tau_n$, with the dominant contribution arising from the integral $\int_{-i\infty}^0 e^{-2i(\tau - \tau_n)t} f(t) dt$, where $f(t) = \zeta(-it) + 2e^{i\pi/4}\sqrt{t}$.
• The amplitude of the solution decays as $\varepsilon / |\tau - \tau_n|$ near $\tau_n$, with a prefactor involving $c_n = e^{i(2\tau_n - 3)/\varepsilon + i\pi/4}$, indicating a non-trivial phase accumulation.
• The leading-order asymptotic expression for the solution near the threshold is given by $\mathcal{G} = \frac{c_n e^{i\pi/4} \varepsilon \sin x}{\sqrt{2\pi}(\tau - \tau_n)} \int_0^\infty e^{-2(\tau - \tau_n)t} \left( \zeta(-it) + 2e^{i\pi/4}\sqrt{t} \right) dt + O(\varepsilon^{2/3}/z^{5/2})$.
• The transition is characterized by a complex interplay between the zeta function and square-root singularities, with the real part of the integral contributing to the oscillatory decay and phase shift.
• The solution remains well-approximated by the adiabatic ansatz (2) until the eigenvalue reaches the continuous spectrum, after which the asymptotic form changes significantly due to the loss of bound state character.
• The analysis confirms that the eigenvalue disappearance is not abrupt but involves a smooth transition governed by the analytic structure of the scattering data and the behavior of $R(p)$ near $p=1$.