On decaying rate of quantum states

The survival amplitude of a quantum state (wave function) under the Schrodinger evolution can be expressed as the Fourier transform of the probability density induced by the wave function in the energy representation. In particular, the first zero of the survival amplitude is a fundamental quantity in characterizing the decaying rate of the quantum state. A basic problem in quantum mechanics is to study how fast the survival amplitude can fall. We present a general estimation of the decaying rate of a quantum state in terms of a moment of any order. The result is established by integrating an inequality which involves controlling trigonometric sums by power functions. This inequality is of independent interest in estimating exponential sums.