Dirac delta-convergence of free-motion time-of-arrival eigenfunctions

TL;DR

The paper formally proves that the Aharonov-Bohm TOA operator for a free particle admits complex-eigenvalued eigenfunctions that evolve unitarily to a collapsed arrival at the arrival point, with the real part of the eigenvalue $\tau_R$ giving the arrival time and the imaginary part $\tau_I$ governing the sharpness and energy uncertainty $\Delta E=\hbar/(2\tau_I)$. By analyzing time evolution and employing a Gel'fand–Shilov delta-sequence criterion, the authors show that the time-evolved position density at $t=\tau_R$ converges to a dirac delta $\delta(q)$ as $\tau_I\to 0^+$, confirming that AB eigenfunctions correspond to states with definite arrival times. They distinguish nodal and nonnodal types, each yielding a different delta sequence yet sharing the same limit support, and discuss the physical interpretation of real-eigenvalued states as idealized (infinite-energy-uncertainty) objects. The results solidify the AB operator as a valid free QTOA observable, illuminate the role of complex eigenvalues outside the Hermitian domain, and relate to the confined-TOA framework and Kijowski’s distribution.

Abstract

Previous numerical analyses on the Aharonov-Bohm (AB) operator representing the quantum time-of-arrival (TOA) observable for the free particle have indicated that its eigenfunctions represent quantum states with definite arrival time at the arrival point. In this paper, we give the mathematical proof that this is indeed the case. An essential element of this proof is the consideration of the eigenfunctions of the AB operator with complex eigenvalues. These eigenfunctions can be considered legitimate TOA eigenfunctions because they evolve unitarily to collapse at the arrival point at the time equal to the real part of their eigenvalue. We show that the time-evolved TOA position probability density distribution evaluated at the time equal to the real part of the eigenvalue forms a dirac delta sequence in the limit as the imaginary part of the eigenvalue approaches zero.

Figures (4)

• Figure 1: Time-evolved probability densities for the eigenfunctions $\varphi_{\tau,0}(q,t)$$(a,b)$ and $\varphi_{\tau,1}(q,t)$$(c,d)$ of the AB operator with $\tau = 0.5 + 0.01i$. The eigenfunctions exhibit unitary collapse with the time at which the collapse is sharpest occurring at $t=\tau_R$.
• Figure 2: Time-evolved probability densities for the eigenfunctions $\varphi_{\tau,0}(q,t)$$(a,b)$ and $\varphi_{\tau,1}(q,t)$$(c,d)$ of the AB operator with $\tau = 0.5 + 0.005i$. As seen from the top view, the peak of the collapse occurred at $t=\tau_R$. The peaks of (a) and (c) are higher than the peaks of (a) and (c) in Figure 1 indicating that the unitary collapse becomes sharper as $\tau_I$ decreases.
• Figure 3: Width-at-half-maximum plot for a.) $|\varphi_{\tau,0}(q,t)|^2$ and b.) $|\varphi_{\tau,1}(q,t)|^2$ for different times $t$ with $\tau = \tau_R + i\tau_I$ where $\tau_R = 0.5$ while $\tau_I$ is varied (see legend). For a given $\tau$, WHM is minimum at $t=\tau_R$. The WHM at $t=\tau_R$ decreases with decreasing $\tau_I$ .
• Figure 4: Concavity of $\sigma_q^{\gamma}(t)$ at $t=\tau_R$ for different values of $\gamma$ and for $\tau_I=1$. The concavity is positive for $0<\gamma<2$ which implies that $\sigma_q^{\gamma}(t)$ has a local maximum at $t=\tau_R$. Note that the we can apply an appropriate change of variable to place the dependence of $\tau_I$ outside the integral in Eq. \ref{['eq:spread-mod']} so that this result holds true for all $\tau_I>0$.

Theorems & Definitions (1)

• Theorem : Gelfand-Shilov