{'fraction_non_alphanumeric': 0.01687289088863892, 'fraction_numerical': 0.006186726659167604, 'mean_word_length': 4.871287128712871, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Localizing gravity on thick branes : a solution for massive KK modes of the Schrödinger equation

We generate scalar thick brane configurations in a 5D Riemannian space time which describes gravity coupled to a self–interacting scalar field. We also show that 4D gravity can be localized on a thick brane which does not necessarily respect Z2–symmetry, generalizing several previous models based on the Randall–Sundrum system and avoiding the restriction to orbifold geometries as well as the introduction of the branes in the action by hand. We begin by obtaining a smooth brane configuration that preserves 4D Poincaré invariance and violates reflection symmetry along the fifth dimension. The extra dimension can have either compact or extended topology, depending on the values of the parameters of the solution. In the non–compact case, our field configuration represents a thick brane with positive energy density centered at y = c2, whereas in the compact case we get pairs of thick branes. We recast as well the wave equations of the transverse traceless modes of the linear fluctuations of the classical solution into a Schrödinger’s equation form with a volcano potential of finite bottom. We solve Schrödinger equation for the massless zero mode m = 0 and obtain a single bound wave function which represents a stable 4D graviton and is free of tachyonic modes with m < 0. We also get a continuum spectrum of Kaluza–Klein (KK) states with m > 0 that are suppressed at y = c2 and turn asymptotically into plane waves. We found a particular case in which the Schrödinger equation can be solved for all m > 0, giving us the opportunity of studying analytically the massive modes of the spectrum of KK excitations, a rare fact when considering thick brane configurations.