Low-temperature anomalies of $^4$He crystals

Low-temperature anomalies of hcp $^4$He crystals (mass decoupling from a torsional oscillator, shear modulus anomaly, dissipation peaks, heat capacity peak) are explained. A simple model based on the concept of resonant tunneling systems in imperfect crystals is proposed. Mass decoupling is caused by an internal Josephson effect: the mass supercurrent inside phase coherent tunneling systems. Quantitative results are in reasonable agreement with experiments.