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Theory of the quantum quartic oscillator is developed with close attention to the cutoff one needs to impose on the system in order to approximate the smallest eigenvalues and corresponding eigenstates of its Hamiltonian by diagonalizing matrices of limited size. The matrices are obtained by evaluating matrix elements of the Hamiltonian between the associated harmonic-oscillator eigenstates and by correcting the computed matrices to compensate for their limited dimension, using the Wilsonian renormalization-group procedure. When the oscillator is used to represent the zero-momentum mode of a scalar quantum field, the cutoff limits the number of quanta one includes in the dynamics, in analogy to the Tamm-Dancoff approach to solving Hamiltonian eigenvalue problems in quantum field theory. The cutoff dependence of the corrected matrices is found to be described by a spiral motion of a three-dimensional vector. This behavior is shown to result from a combination of a limit-cycle and a floating fixed-point behaviors, an unexpected phenomenon that warrants further study for the purpose of learning how the limits on the number of quanta can be accounted for in more complex systems. A brief discussion of the research directions concerning polynomial interactions of degree higher than four, spontaneous symmetry breaking and coupling of more than one oscillator, is included.

This study examines the modal properties of fractal-idealized monopodial trees, which comprise a main trunk with branches grown laterally and axially from it. Analysis via a renormalization technique shows that monopodial trees possess emerging modes , self-similar modes with lateral branching , and self-similar modes with axial branching. The study introduces a recursive analytical approach, which involves the construction of auxiliary P -functions to characterize the modal properties. Results reveal that monopodial trees' modal frequencies are closely spaced because their emerging modes increase exponentially in number. Under the self-similar modes with lateral branching, the trees inherit all modes from their fractal ancestors, while under the self-similar modes with axial branching, they inherit only the self-similar modes from their ancestors. Hence, the main trunk stands still at self-similar modes, and the trees localize vibration at higher-order lateral branches. Therefore, the monopodial branching architecture is advantageous for reducing the vibration of the main trunk. This study also derives analytical formulas for the modal frequencies of trees with infinite order via a group tree modeling approach. Monopodial trees acquire the highest natural frequency when they have a one-to-one lateral-to-axial-branching ratio. The proposed formulas are verified with independent literature results and are shown to be accurate. • This study examines the modal properties of fractal monopodial trees. • Monopodial trees have emerging modes and two distinct self-similar modes. • Auxiliary P -functions construct trees' modal properties recursively. • Modal properties of infinite-order trees are derived using group tree modeling. • One-to-one lateral-to-axial-branching ratios yield the highest natural frequency. [ABSTRACT FROM AUTHOR]