Following our work [Phys. Rev. Lett. 125, 020401 (2020)], we discuss a semiclassical description of one-dimensional quantum tunneling through multibarrier potentials in terms of complex time. We start by defining a complex-extended continuum level density of unbound systems and show its relation to a complex time shift of the transmitted wave. While the real part of the level density and time shift describes the passage of the particle through classically allowed coordinate regions, the imaginary part is connected with an instantonlike picture of the tunneling through forbidden regions. We describe singularities in the real and imaginary parts of the level density and time shift caused by stationary points of the tunneling potential, and show that they represent a dual extension of excited-state quantum phase transitions from bound to continuum systems. Using the complex scaling method, we numerically verify the predicted effects in several tunneling potentials.

中文翻译：

---

激发态量子相变的连续介质类似物

按照我们的工作 [ Phys. 牧师莱特。 125, 020401 (2020)]，我们讨论了在复杂时间方面通过多势垒势的一维量子隧穿的半经典描述。我们首先定义未绑定系统的复数扩展连续介质能级密度，并展示其与传输波复数时移的关系。虽然能级密度和时移的实部描述了粒子通过经典允许坐标区域的通道，但虚部与通过禁止区域的隧道的瞬时图像相关联。我们描述了由隧道势的驻点引起的能级密度和时移的实部和虚部中的奇点，并表明它们代表了激发态量子相变从束缚系统到连续系统的双重扩展。使用复杂的缩放方法，