Due to the complexity of socio-economic-related issues, people thought of housing market as a chaotic nucleus situated at the intersection of neighboring sciences. It has been known that the dependence of house features on the residential property value can be estimated employing the well-established hedonic regression analysis method in teams of location characteristic, neighborhood characteristic and structure characteristic. However, to further assess the roles of urban infrastructures in housing markets, we proposed a new kind of volatility measure for house prices utilizing the Lie symmetry analysis of quantum theory based on Schrödinger equation, mainly focusing on the effects of transportation systems and public parks on residential property values. Based on the municipal open government data regularly collected for four cities, including Boston, Milwaukee, Taipei and Tokyo, and all spatial sampling sites were featured by United States Geological Survey (USGS) National Map, transportation and park were modelled as perturbations to the quantum states generated by the feature space in response to the environmental amenities with different spatial extents. In an attempt to ascertain the intrinsic impact of the location-dependent price information obtained, the similarity functions associated with the Schrödinger equation were considered to facilitate revealing the city amenities capitalizing into house prices. By examining the spatial spillover phenomena of house prices in the four cities investigated, it was found that the mass transit systems and the public green lands possessed the infinitesimal generators of Lie point symmetries Y2 and Y5, respectively. Compared statistically with the common performance criteria, including mean absolute error (MAE), mean squared error (MSE) and, root mean squared error (RMSE) obtained by hedonic pricing model, the Lie symmetry analysis of the Schrödinger equation approach developed herein was successfully carried out. The invariant-theoretical characterizations of economics-related phenomena are consonant with the observed residential property values of the cities internationally, ultimately leading to develop a new perspective in the global financial architecture.

The invariant metrics of the effects of park size and distance to public transportation on housing value volatilities in Boston, Milwaukee, Taipei and Tokyo are investigated. They reveal a Cobb-Douglas-like behavior. The scaleinvariant exponents corresponding to the percentage of a green area (a) are 7.4, 8.41, 14.1 and 15.5 for Boston, Milwaukee, Taipei and Tokyo, respectively, while the corresponding direct distances to the nearest metro station (d) are −5, −5.88, −10 and −10, for Boston, Milwaukee, Taipei and Tokyo, respectively. The multiphysics-based analysis provides a powerful approach for the symmetry characterization of market engineering. The scaling exponent ratio between park area percentages and distances to metro stations is approximately 3/2. The scaling exponent ratio expressed in the perceptual stimuli will remain invariant under group transformation. According to Stevens' power law, the perception-dependent feature spaces for parks and public transportation can be described as two-and three-dimensional conceptual spaces. Based on the prolongation structure of the Schrödinger equation, the SL(2, R) models are used to analyze the house-price volatilities. Consistent with Shepard's law, the rotational group leads to a Gaussian pattern, exhibiting an extension of the special linear group structure by embedding SO(3) ⊗ R(3) in SL(2, R). The influencing factors related to cognitive functioning exhibit substantially different scaleinvariant characteristics corresponding to the complexity of the socio-economic features. Accordingly, the contour shapes of the price volatilities obtained from the group-theoretical analysis not only corroborate the impact of the housing pricing estimation in these cities but also reveal the invariant features of their housing markets are faced with the forthcoming sustainable development of big data technologies and computational urban science research.