Crossover between quantum and classical waves and high-frequency localization landscapes

Anderson localization is a universal interference phenomenon occurring when a wave evolves through a random medium and it has been observed in a great variety of physical systems, either quantum or classical. The recently developed localization landscape theory offers a computationally affordable way to obtain useful information on localized modes, such as their location or size. Here we examine this theory in the context of classical waves exhibiting high-frequency localization and for which the original localization landscape approach is no longer informative. Using the so-called Webster's transformation, to convert a classical wave equation into a Schrödinger equation with the same localization properties, and combining a set of frequency-shifted operators, we introduce an optimized localization landscape. This optimized localization landscape offers an affordable way to reveal key information on mode localization across the frequency spectrum.