Existence and number of real spectral roots

Determine the existence and number of real eigenvalue roots of the transcendental spectral equation governing the Maxwell–Cattaneo–Vernotte problem with one adiabatic and one heat-transfer boundary condition for arbitrary dimensionless relaxation time and Biot number.

Background

The eigenvalues are obtained from a transcendental equation whose roots may be real, purely imaginary, or complex. The paper provides numerical procedures for locating real roots by examining the extrema of the real-variable form of the equation.

However, the authors explicitly state that neither the existence nor the number of real roots is known in general. Thus, a complete analytical characterization of the real spectrum as a function of the dimensionless relaxation time and Biot number remains unresolved, despite the numerical root-finding procedure developed in the appendix.

References

Since neither the existence nor the number of real eigenvalue roots is known, the transcendental equation \re{ftp:omega} and its derivative \wrt $ \omega $ are investigated simultaneously.

— Exact analytical solution for the non-selfadjoint problem of Maxwell--Cattaneo--Vernotte heat conduction with heat-transfer boundary condition  (2608.13164 - Szücs et al., 13 Aug 2026) in Appendix A, subsection “Real eigenvalues”