Winding in non-Hermitian systems

Abstract This paper extends the property of interlacing of the zeros of eigenfunctions in Hermitian systems to the topological property of winding number in non-Hermitian systems. Just as the number of nodes of each eigenfunction in a self-adjoint Sturm–Liouville problem are well-ordered, so too are the winding numbers of each eigenfunction of Hermitian and of unbroken <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mrow> <mml:mi mathvariant="script">P</mml:mi> <mml:mi mathvariant="script">T</mml:mi> </mml:mrow> </mml:mstyle> </mml:math> -symmetric potentials. Varying a system back and forth past an exceptional point changes the windings of its eigenfunctions in a specific manner. Nonlinear, higher-dimensional, and general non-Hermitian systems also exhibit manifestations of these characteristics.

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