![]() Therefore, Equations (2) and (3) are fulfilled for components of v, since the corresponding equations for v components are complex conjugates of, at most, a rearrangement of their u-component counterparts. Then, for every integer j in the interval, v 2 j − 1 = u ¯ 2 j and v 2 j = u ¯ 2 j − 1 and, for every integer j in the interval, v j = u ¯ j. ![]() Expressed in words, P is the operator that interchanges the (2 j − 1)th and (2 j)th (1 ⩽ j ⩽ m) components of the N-tuple it is acting on and leaves the rest (if any) of the components unchanged.įirst, we show that Dom( L h) is invariant under the action of PT. We now show that PT, where P ≔ ∏ j = 1 m P 2 j − 1, 2 j, is a symmetry of L h. Without loss of generality, one can assume that the components of h are coming in an order in which h 2 j = h ¯ 2 j − 1, for all integers 1 ⩽ j ⩽ m, and all other components (if any) are real. Let 2 m, for some integer m where 1 ⩽ m ⩽ ⌊ N/2⌋, be the number of non-real components of h. Therefore, suppose that this is not the case and, since h is assumed to be invariant under conjugation, suppose that h has at least two non-real components. If all components of h are real, then it is straightforward to see that, say, P 1, 1 T = i d ℋ T is a symmetry of the quantum graph. Our main goal is to prove the following theorem. Such symmetry operator is defined using the eigenfunctions and there is no guarantee that it comes from an automorphism of the metric graph. For an operator with discrete spectrum and eigenfunctions of the operator and its adjoint building a biorthogonal basis, one may easily construct a symmetry operator in the Hilbert space, provided the spectrum is reflection symmetric. 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. ![]() (x,y)\rightarrow (−y,−x)\).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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