RT-Symmetric Laplace Operators on Star Graphs : Real Spectrum and Self-Adjointness

How ideas of PT-symmetric quantum mechanics can be applied to quantum graphs is analyzed, in particular to the star graph. The class of rotationally symmetric vertex conditions is analyzed. It is shown that all such conditions can effectively be described by circulant matrices: real in the case of odd number of edges and complex having particular block structure in the even case. Spectral properties of the corresponding operators are discussed.


Introduction
Writing this paper we got inspiration from two rapidly developing areas of modern mathematical physics: PTsymmetric quantum mechanics and quantum graphs.Both areas attract interest of both mathematicians and physicists for the last two decades with numerous conferences organized and articles published.The first area grew up from the simple observation that a quantum mechanical Hamiltonian "often" has real spectrum even if it possesses combined parity and time-reversal symmetry instead of self-adjointness [1][2][3][4][5][6].Considering such operators increases the set of physical phenomena that could be modeled and raises up new interesting mathematical questions [7][8][9][10].It appears that spectral theory of such PT-symmetric operators (see precise definition below) can be well-understood using framework of self-adjoint operators in Krein spaces [11].
The theory of quantum graphs-differential operators on metric graphs-can be used to model quantum or acoustic systems where motion is confined to a neighborhood of a set of (one-dimensional) intervals [12][13][14].Until now quantum graphs were mostly studied in the context of self-adjoint or dissipative operators.Our key idea is to look at differential operators on metric graphs under more general symmetry assumptions reminding those in PT-symmetric theory.Surprisingly spectral properties of operators on graphs with symmetries have not been paid much attention.We mention here just two papers [15,16], where symmetries of graphs were used to construct counterexamples showing that inverse problems are not necessarily uniquely solvable.
In quantum graphs motion along the edges is described by ordinary differential equations, which are coupled together by certain vertex conditions connecting together values of the functions at the end points of the intervals building the underlined metric graph.The role of vertex conditions is twofold: to describe how the waves are penetrating through the vertices and to make the differential operator self-adjoint.If the requirement of self-adjointness is waved then such conditions should instead ensure that the resolvent set is not empty; that is, the resolvent for the corresponding differential operator exists for some .The later condition is not very precise and one of the goals of the current paper is to understand it in the case of the simplest merit graph with symmetries-the star graph.It can be considered as a building block to define differential operators on arbitrary metric graphs.To avoid discussing properties of the differential operator we limit our studies to the Laplace operator.Moreover the graph formed by semi-infinite edges is considered in order to avoid influence from the peripheral vertices.We just mention here that star graphs formed by finite edges but with standard conditions (see ( 2)) at the central vertex were considered recently in the framework of

Notations and Elementary Properties
Our goal is to generalize ideas that originated from PTsymmetry for the case of the star graph.More precisely we will confine our studies to the case of the Laplace operator with the domain given by generalized Robin conditions at the central vertex.Consider the star graph Γ  formed by  semi-infinite edges   = [0, ∞) joined together at one central vertex.The corresponding Hilbert space  2 (Γ  ) can be identified with the space of vector valued functions  ≡ ⃗ () on  ∈ [0,∞) with the values in Definition 1.The operator   = − 2 / 2 is defined on the set of functions from the Sobolev space  2  2 ([0, ∞); C  ) satisfying generalized Robin conditions: where  is a certain  ×  matrix.
The operator   can be seen as a certain point perturbation of the self-adjoint standard Laplace operator defined on the domain of functions from the same Sobolev space satisfying standard vertex conditions: In fact all such point perturbations are defined by vertex conditions of a more general form (8), but Theorem 2 implies that only conditions of form (1) are important for our goal.
The operator adjoint to   is again the Laplace operator but is defined by vertex conditions (1) with the matrix  substituted by the matrix  * that is the operator   * : This can be proven by integration by parts for ⃗  ∈ Dom(  ), ⃗ V ∈ Dom( *  ): where we used the fact that ⃗  satisfies (1).This formula defines a bounded functional with respect to ⃗  if and only if It follows that the operator   is self-adjoint if and only if  is a Hermitian matrix  * = .In this paper we are not interested in the case where   is self-adjoint.
The spectrum of the operator   may contain up to  isolated eigenvalues.The corresponding eigenfunction is a solution to the differential equation satisfying Robin conditions (1).Any square integrable solution to the differential equation is given by with Im  > 0. This function satisfies Robin condition if and only if The last equation has at most  distinct solutions (in the correct half-plane).Observe that not all solutions lead to eigenfunctions, since one needs to meet the condition Im  > 0.
As the theory of PT-symmetric operators indicates the most interesting case is when the spectrum of the operator is pure real, the operator itself is not self-adjoint.We are going to look closer at such operators.If the operator   has  real eigenvalues, then the matrix  has  negative eigenvalues, but it does not imply that it is Hermitian.
Note that possible vertex conditions are not limited to those described by (1).More generally one may consider the Laplace operator  , = − 2 / 2 defined on the domain of functions from  2  2 ([0, ∞); C  ) satisfying the following vertex conditions: where ,  are certain × matrices, such that rank (, ) = .Here we follow ideas from [22].The following theorem shows that a Laplace operator described by the vertex conditions of the form (8) has  real eigenvalues if either the vertex conditions are of the form (1) or the spectrum covers the whole complex plane.
Theorem 2. Consider the operator  , defined by vertex condition (8).If  , has  real eigenvalues (counting multiplicities), then either  is invertible or the spectrum of  , is the whole complex plane.
Proof.A function  is an eigenfunction of  , if and only if it satisfies the differential equation ( 5), which has solution (6).Substituting in (8), we get that for a certain ⃗  ̸ = ⃗ 0 Conversely, if this holds for some  with Im  > 0 then  2 is an eigenvalue of  , .
The last equation has nontrivial solution if and only if Let us now consider two invertible matrices  and .If ( 10) is satisfied then also Let us take  and  such that Let us introduce the following function: () =: det( + ).Because of ( 12),  is a polynomial in  of degree at most rank .If  ≡ 0, then any  with Im  > 0 gives an eigenvalue  2 and as a result the spectrum is the whole complex plane.Otherwise  has at most rank  zeros.Suppose now that each eigenvalue has multiplicity 1, then if rank  ̸ = , the operator will have less than  eigenvalues.Therefore we must have the fact that  is invertible.To complete the proof, the case of multiple eigenvalues has to be considered.Let us assume that  2 0 is an eigenvalue of multiplicity .To see that  must be invertible if  , has  eigenvalues, it is enough to prove that () has a zero of order at least  at  0 .To prove this, we consider the eigenfunctions has a zero of order at least  at  =  0 .
Our method to obtain nontrivial operators with nonstandard symmetries is a certain generalization of the method of point interactions originally developed in the framework of self-adjoint Hamiltonians [23,24].The phenomenon described in Theorem 2 in connection with PT-symmetric point interaction was first observed in [25], following [26].
This theorem implies that the class of operators   defined by Robin vertex conditions (1) is rather wide; therefore in what follows we focus our attention on this class only.

Pseudo-Hermitian and Pseudoreal Operators
Our studies are inspired by recent papers devoted to investigation of the so-called PT-symmetric operators in one dimension.An operator  is called PT-symmetric if it satisfies the following relation: where P is the spacial symmetry operator (parity symmetry), and T is the antilinear operator of complex conjugation (time-reversal symmetry), PT-symmetry (like usual operator symmetry) is not enough to guarantee that the corresponding operator is physically relevant: it might happen that its spectrum is empty (take, e.g., the second derivative operator on the interval [−, ] with both Dirichlet and Neumann conditions imposed on both end points of the interval).In conventional quantum theory the notion of a self-adjoint operator substitutes simple symmetry property (of course any self-adjoint operator is symmetric, but not the other way around).Therefore it appears natural to substitute relation ( 14) with the following one: where we use the fact that P is unitary P −1 = P * .In the case of conventional PT-symmetric theory the operator P is not only unitary, but also self-adjoint P * = P.It follows that the operator satisfying ( 17) is pseudo-self-adjoint; that is, it is self-adjoint not in the original Hilbert space but in the Krein space with the sesquilinear form [⋅, ⋅] defined by P as Gram operator: where ⟨⋅, ⋅⟩ denotes the standard scalar product in  2 .
The goal of the current paper is to generalize PTsymmetry for the case of operators on metric graphs, more precisely for the star graph Γ  .Operator on such a star graph can be considered as a building block to define operators on arbitrary graphs.We are going to substitute the operator of We are not going to distinguish the rotation operators in C  and in the space of vector-valued functions hoping that this will not lead to any misunderstanding.Observe that the operator R is unitary but not self-adjoint (of course provided  ̸ = 2).
We would like to understand whether ideas from PTsymmetric theory may lead to an interesting new class of operators   , which are not self-adjoint and not PTsymmetric with a suitably defined self-adjoint spacial symmetry operator P. For example, if  is even then such a spacial symmetry operator can be defined as Using P we may introduce PT-symmetric operators on Γ  , but this would be just a vector version of conventional onedimensional theory (see Lemma 4 below).Formulas ( 14) and ( 17) suggest that we look closer at the following two possible generalizations of the notion of PTsymmetry: (ii) pseudo-Hermitian operators We are going to reserve the term pseudo-self-adjoint for operators which are pseudo-Hermitian with respect to a selfadjoint operator P as in (17).Surprisingly pseudo-Hermitian operators do not define any new interesting class as follows from the two lemmas below.

Lemma 3. Let 𝑁 be an odd number; then the operator 𝐿
Proof.Iterating formula (22) one gets the following set of equations: Taking into account that R  is the identity operator we arrive at the following relation for any odd : proving our statement.

Lemma 4.
Let  be an even number.
(i) If in addition /2 is an odd number, then the operator   is R-pseudo-Hermitian only if it is pseudo-selfadjoint with respect to P = R /2 .
(ii) If in addition /2 is an even number, then the operator   is R-pseudo-Hermitian only if it commutes with the self-adjoint rotation P = R /2 and therefore is unitarily equivalent to an orthogonal sum of Laplace operators on Γ /2 with Robin conditions at the central vertices.
The operators  ± appearing in the previous Lemma satisfy symmetry properties similar to (22), but with the "rotation" operators R ± of lower size (/2 instead of ).If R + is the standard rotation operator in the space C /2 , the operator R − is a certain modified rotation operator: It might be interesting to understand the symmetry of  − in more detail, but we may conclude already now that in most cases the operator   is pseudo-Hermitian only if it is also pseudo-self-adjoint (with respect to another symmetry operator).Therefore in what follows we focus on the studies of pseudo-real realisations of the Laplace operator on the star graph Γ  .Therefore we are going to use the following definition.
Definition 5.An operator  in  2 (Γ  ) is called RTsymmetric if and only if it satisfies the following relation: where T is the antilinear operator of complex conjugation.
This definition guarantees that the spectrum of the operator is symmetric with respect to the real axis.Really if () is an eigenfunction corresponding to the eigenvalue , then () = RT is also an eigenfunction but corresponding to the eigenvalue , provided (27) holds: Hence with Definition 5 we always have spectrum which is symmetric with respect to the real axis as in the classical PTsymmetric theory.

RT-Symmetry of Point Interactions
In the current section we are going to describe the structure of RT-symmetric operators.It appears that the corresponding matrices  belong to the class of circulant matrices which we describe now.A circulant matrix is a special case of a Toeplitz matrix.Definition 6.An  ×  matrix  = {  } is called circulant if the value of the entry   depends only on the difference ( − ) mod ; that is, , where   ,  = 0, . . .,  − 1, are block matrices of the same size  × ,  = .
The following theorem describes matrices  leading to RT-symmetric operators on Γ  .

Theorem 8. Consider the operator 𝐿 𝐴 determined by Definition 1.
If  is odd, then the operator   is RT-symmetric if and only if  is a real circulant matrix; that is,  = circ ( 0 ,  1 , . . .,  −1 ) ,   ∈ R,  = 0, . . .,  − 1. ( If  is even, then the operator   is RT-symmetric if and only if  is a complex /2×/2 block circulant matrix formed by the following 2 × 2 blocks: ) , Proof.Let us consider the operators   as defined in Definition 1. Suppose that the function  ∈ Dom(  ) and therefore satisfies the boundary condition (1).The boundary condition for the function RT is given by This condition should be identical with (1) leading to where we used the fact that the rotation matrix R has real entries.
Let us denote the entries of the matrix  by  , ; then the last equality implies The structure of the matrix  is as follows: every next row in the matrix is equal to the previous one shifted to the right one step and conjugated.It is clear then that  is determined by  complex numbers, for example, those building the first row.If there would be no complex conjugation or the entries would be real, then  would be circulant.Now, we consider the cases when  is odd and even separately.

Advances in Mathematical Physics
Therefore and we see that the matrix  is a real circulant matrix.It is determined by  real parameters: Is Even.We again consider (34) to obtain the following: We do not obtain any restriction on  , and hence the entries of the first row can be chosen arbitrarily among complex numbers.But we still have the following property, which reminds us of circulant matrices: provided   = ( + 2) mod  and   = ( + 2) mod ,  = 1, . . ., /2.Let us denote the first row in  by   ,  = 0, . . ., −1; then each row of the matrix  is the conjugation of the previous row shifted to the right; that is, as claimed.We see that  is a 2×2 block circulant matrix.The blocks forming  are not chosen arbitrarily but depend on  complex parameters.

On the Spectrum of the RT-Symmetric Operators
We can now look at the discrete spectra of the constructed RT-symmetric operators on the star graph Γ  .We have already observed that, as in the case of all PT-symmetric operators, the nonreal eigenvalues of a RT-symmetric operator always appear in conjugate pairs (28).That is, if  is an eigenvalue of a RT-symmetric operator A, then  is also an eigenvalue of the operator A. We noted that the operator   may have at most  distinct eigenvalues (counting multiplicities).The most interesting case is when the spectrum of the operator is real; also the operator itself is not self-adjoint.The eigenvalues of   are given by solutions of ( 7) with Im  > 0. Negative eigenvalues correspond to  on the upper part of the imaginary axis.In what follows we study the case where the operator has precisely  (negative) real eigenvalues.Since the structures of the matrices are different in the cases when the number of edges  is even or odd, these cases will be studied separately.

An Odd Number of Edges.
Before we study the discrete spectrum of the operator   , we recall some known results about the eigenvalues of a circulant matrix which can be nicely calculated as follows [27].Proposition 9. Let  = circ( 0 ,  1 , . . .,  −1 ) be a circulant matrix; then its eigenvalues are given by 2/ ,  = 0, . . .,  − 1. (42) Proof.The key idea is to write arbitrary circulant matrix  as a sum of powers of the rotation matrix R. The rotation matrix R can be seen as an elementary circulant matrix: R = circ (0, 1, 0, . . ., 0) .
Of course, if the number of real eigenvalues is less than  the operator   does not need to be self-adjoint.The proof of the later theorem may give an impression that whether the size of  is even or odd does not play any essential role.The next subsection shows that the difference is tremendous.

An Even Number of Edges.
We recall from Theorem 8 that in case the star graph Γ  has even number of edges the operator   is RT-symmetric if and only if  = circ( 0 ,  1 , . . .,   ), where  = /2, and A vector ⃗  ̸ = 0 is an eigenvector of  with eigenvalue  if and only if Following the ideas in [28], we will look for eigenvectors of  of the following form (this is a natural suggestion given the structure of the eigenvectors for the case when  is odd): where V is a nonzero two-dimensional vector and  is a fixed th root of the unity; that is,  ∈ { (2/) } ,  = 0, . . .,  − 1.
One may prove that all eigenvectors of  are of this form, since  is commuting with R 2 .
Extending (50) with  = circ( 0 ,  1 , . . .,   ), as explained above, the following set of  equations is obtained: (55) Each eigenvector V = V() of () will generate an eigenvector ⃗  of  with the same eigenvalue.If for every  the 2 × 2 matrix () has two negative eigenvalues, then the matrix  has precisely  negative eigenvalues and so the operator   .
The following theorem is a counterpart of Theorem 10 for the case when  is even.Theorem 11.Let the number  of edges of the star-graph Γ  be even.Then among RT-symmetric operators   (given by Definition 1) there are some with  real eigenvalues that are not self-adjoint.Proof.To prove the theorem it is enough to present an example of such a matrix  for arbitrary even .It is enough to find  such that all () have two negative eigenvalues.As by Theorem 8 if  is even then   is RT-symmetric if and only if  is block circulant.Consider such block circulant matrix  given by Advances in Mathematical Physics spacial symmetry P with the rotation operator R defined as follows:
Proof.First, we note that  < 0 is an eigenvalue of   if and only if  = − √ − is an eigenvalue of the matrix .If  is odd, then in accordance with Theorem 8  is a circulant matrix with real entries.Then its eigenvalues are nothing else other than a discrete Fourier transform of {  } −1 =0