Avoided Crossing

An avoided crossing is the approach and subsequent separation of two eigenvalues of a matrix or linear operator family without their becoming equal as a real parameter varies. It is often called an avoided level crossing.

For example, consider the real symmetric matrix family

where and is a fixed nonzero real number. Its eigenvalues are

The gap has minimum at . When , the two diagonal entries and instead cross at the origin.

For a general real symmetric matrix of size , a repeated eigenvalue requires both and . These are two independent real conditions, so the coincidence has codimension 2. For a Hermitian matrix, the off-diagonal entry may be complex, and its vanishing imposes two real conditions in addition to , giving codimension 3 (von Neumann and Wigner 1929; Seyranian et al. 2005, p. 1725). Thus an unconstrained family with one real parameter generically avoids such coincidences within either of these matrix classes. Additional symmetry can permit an actual crossing.

For fixed , analytic continuation of the displayed family to complex gives square-root branch points at , where is the imaginary unit. At either value the matrix is nonzero but its square vanishes, so the coincident eigenvalues have only one independent eigenvector. These are exceptional points. This model illustrates the connection between avoided crossings for a real parameter and branch point structure in the complex plane (Heiss and Steeb 1991). A diabolic point, in contrast, has two linearly independent eigenvectors at a double eigenvalue.