Isoscattering strings of concatenating graphs and networks.

Area of Science:

• Quantum physics
• Network theory
• Mathematical physics

Background:

• Isospectrality and isoscattering are crucial in quantum systems.
• Understanding the relationship between network shape and scattering properties is challenging.
• Previous methods for analyzing complex networks are computationally intensive.

Purpose of the Study:

• To identify and investigate isoscattering properties of concatenating quantum graphs.
• To develop principles for designing large, isoscattering graphs and networks.
• To experimentally validate theoretical predictions for quantum graph networks.

Main Methods:

• Theoretical analysis of isoscattering strings of quantum graphs.
• Experimental verification using microwave networks with multiple leads.
• Application of a novel trace function to simplify scattering matrix analysis.

Main Results:

• Demonstrated principles for designing isoscattering quantum graphs.
• Confirmed theoretical predictions experimentally with microwave networks.
• Showcased the trace function's ability to reduce scattering matrix complexity from all entries to 2n diagonal elements.

Conclusions:

• The trace function provides a new method to determine if scattering properties uniquely connect to the shape of open complex networks.
• This work extends the 'Can one hear the shape of a drum?' problem to open quantum graphs and networks.
• The findings offer insights into designing complex networks with specific scattering characteristics.