Reconstructing the Sanxian's Music Score by a Birth--Death Matrix

The Sanxian is a traditional Chinese three-stringed plucked instrument. Its music can be generated by tridiagonal complex matrices. The sound people hear is determined by its spectrum and naturally requires that the matrix has a real spectrum. As in quantum mechanics, the description of the model is a complex operator and the observable measurement is real. In other words, the tridiagonal complex matrix described is a self-adjoint operator on the complex inner product space with respect to a measure. It is well known that the birth--death Q matrix can be matched and naturally self-adjointed. We will introduce the latest representative results: for a fairly wide range of self-adjoint tridiagonal complex matrices, a birth--death Q matrix can always be constructed to make both isospectral (in simple words, both have the same eigenvalues). This problem is simple and easy to understand. But we have studied it from three different perspectives: probability theory, statistical physics and computational mathematics at different times, and have gone through a long time of exploration.