![]() ![]() Our main question is whether the opposite statement holds, namely, whether the reflection symmetry of the spectrum implies P T-symmetry of the operator with respect to a certain automorphism P of the metric graph Γ. Then the spectrum possesses reflection symmetry with respect to the real axis. ![]() It is relatively easy to see that if the set of Robin parameters is invariant under conjugation, then the corresponding Laplace operator is P T-symmetric with respect to a certain automorphism P of the underlying metric graph Γ. One may break the self-adjointness by introducing Robin conditions with non-real parameters at the degree-one vertices. If the so-called standard vertex conditions (continuity of the function and vanishing of the sum of normal derivatives) are introduced, then the corresponding operator is self-adjoint and the spectrum is real (an infinite set of discrete eigenvalues tending to +∞). The metric graph Γ has a rich symmetry group generated by the permutations of the edges. We consider the case of an equilateral star-graph Γ, as the one in Figure 1, formed by N identical edges joined at the central vertex, together with the Laplace operator acting on it. We would like to understand whether this mechanism is unavoidable for a quantum graph to have a reflection-symmetric spectrum. ) If a metric graph possesses a certain automorphism (symmetry) P, then the corresponding differential operator can be chosen to have P T-symmetry, leading to reflection-symmetric spectrum. (In the classical studies P is the reflection operator ( P f ) ( x ) = f ( − x ) and T is the time-reversal operator of complex conjugation ( T f ) ( x ) = f ( x ) ¯. Extending the set of allowed operators by including P T-symmetric ones leads to the spectrum with reflection symmetry with respect to the real axis, not only as a set but also including multiplicities. 3–5,15,16 Standard quantum mechanics in one dimension is described by self-adjoint differential operators leading to purely real spectrum. While some causes can be self-cured at home, a few others need proper treatment from a medical practitioner.The main goal of our current paper is to understand connections to the theory of P T-symmetric operators-yet another area of mathematical physics that has got a lot of attention recently.
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