Analytic eigenvalue structure of a coupled-oscillator system beyond the ground state

Abstract

By analytically continuing the eigenvalue problem of a system of two coupled harmonic oscillators in the complex coupling constant g, we have found a continuation structure through which the conventional ground state of the decoupled system is connected to three other lower unconventional ground states that describe the different combinations of the two constituent oscillators, taking all possible spectral phases of these oscillators into account [7]. In this work we calculate the connecting structures for the higher excitation states of the system and argue that -in contrast to the fourfold Riemann surface identified for the ground state -the general structure is eight-fold instead. Furthermore we show that this structure in principle remains valid for equal oscillator frequencies as well and comment on the similarity of the connection structure to that of the single complex harmonic oscillator.

The Analytic Eigenvalue Structure of the 1+1 Dirac Oscillator

“…Let us begin with the nonrelativistic harmonic oscillator, whose analytical properties have been discussed in [6][7][8]. We give its eigenfunctions in both the conventional and unconventional spectra, based on which one can solve the eigenvalue problem for the relativistic case concisely.…”

“…Denoting the eigenenergies of the Dirac oscillator as E, the eigenvalues of H 2 are E 2 . Corresponding to each solutions in (7), one can obtain…”

“…Experimental studies of exceptional points in several systems [1][2][3][4][5] make the functions of a complex variable being no longer limited to an abstract tool in physics. On the other hand, very recently, the theoretical research on the structure of the Riemann surface has been extended to a coupled quantum system, and unexpected behaviors have been found [6,7]. Namely, in the system of two coupled harmonic oscillators, there exists an eightfold Riemann surface structure of eigenvalues as functions of the complex coupling parameter, which collapses to a fourfold one in the case of the ground state.…”

“…On the one hand, the one-dimensional Dirac oscillator can be derived by linearizing the quadratic form E 2 = m 2 + p 2 + m 2 ω 2 x 2 − βmω, as Dirac's original approach to propose his equation [11], with ω being the frequency and β being one of the Dirac matrices. Hence, it naturally has the property of a square root, which is a key cause of the structure of Riemann surface in the nonrelativistic harmonic systems [6][7][8]. On the other hand, it can be regarded as a coupled quantum system composed by a harmonic oscillator and a spin.…”

“…On the other hand, it can be regarded as a coupled quantum system composed by a harmonic oscillator and a spin. It is interesting to reexamine the Riemann surface when one of the harmonic oscillators in the model stuedied in [6,7] is replaced by a spin.…”