Quantum mechanical potentials related to the prime numbers and Riemann zeros

Area of Science:

• Number Theory
• Quantum Mechanics
• Mathematical Physics

Background:

• The distribution of prime numbers remains a fundamental challenge in mathematics.
• The Riemann zeta function's nontrivial zeros are conjectured to be related to quantum system eigenvalues (Hilbert-Pólya conjecture).
• Connections between number theory and quantum physics are actively explored.

Purpose of the Study:

• To construct quantum potentials whose energy eigenvalues correspond to prime numbers.
• To construct quantum potentials whose energy eigenvalues correspond to the nontrivial zeros of the Riemann zeta function.
• To analyze the mathematical properties of these constructed potentials.

Main Methods:

• Application of the Marchenko inverse scattering approach.
• Construction of specific potentials based on prime number sequences.
• Construction of potentials based on Riemann zeta function zero distributions.
• Analysis of potential properties using multifractal measures (Rényi dimension).

Main Results:

• Successfully constructed potentials with energy eigenvalues matching prime numbers.
• Successfully constructed potentials with energy eigenvalues matching Riemann zeta function zeros.
• Demonstrated that these potentials exhibit multifractal characteristics.
• Quantified the multifractality using the Rényi dimension.

Conclusions:

• The Marchenko approach provides a method to link number theory and quantum mechanics through potential construction.
• The identified multifractal nature of these potentials suggests complex underlying structures.
• These findings offer potential for further analytical and theoretical advancements in both fields.