The Berry-Keating operator on $L^2(\rz_>,\ud x)$ and on compact quantum graphs with general self-adjoint realizations
aa r X i v : . [ m a t h - ph ] M a y The Berry-Keating operator on L ( R > , d x ) and oncompact quantum graphs with general self-adjointrealizations Sebastian Endres
Institut für Theoretische Physik, Universität UlmAlbert-Einstein-Allee 11, 89081 Ulm, GermanyE-mail: [email protected]
Frank Steiner
Institut für Theoretische Physik, Universität UlmAlbert-Einstein-Allee 11, 89081 Ulm, GermanyE-mail: [email protected]
Abstract.
The Berry-Keating operator H BK := − i ~ (cid:0) x dd x + (cid:1) [M. V. Berry andJ. P. Keating, SIAM Rev. 41 (1999) 236] governing the Schrödinger dynamics isdiscussed in the Hilbert space L ( R > , d x ) and on compact quantum graphs. It isproved that the spectrum of H BK defined on L ( R > , d x ) is purely continuous andthus this quantization of H BK cannot yield the hypothetical Hilbert-Polya operatorpossessing as eigenvalues the nontrivial zeros of the Riemann zeta function. A completeclassification of all self-adjoint extensions of H BK acting on compact quantum graphs isgiven together with the corresponding secular equation in form of a determinant whosezeros determine the discrete spectrum of H BK . In addition, an exact trace formula andthe Weyl asymptotics of the eigenvalue counting function are derived. Furthermore,we introduce the “squared” Berry-Keating operator H := − x d x − x dd x − whichis a special case of the Black-Scholes operator used in financial theory of option pricing.Again, all self-adjoint extensions, the corresponding secular equation, the trace formulaand the Weyl asymptotics are derived for H on compact quantum graphs. Whilethe spectra of both H BK and H on any compact quantum graph are discrete, theirWeyl asymptotics demonstrate that neither H BK nor H can yield as eigenvalues thenontrivial Riemann zeros. Some simple examples are worked out in detail.PACS numbers: 03.65.Ca, 03.65.Db Submitted to:
J. Phys. A: Math. Gen. he Berry-Keating operator on L ( R > , d x ) and on compact graphs
1. Introduction: The hypothetical Hilbert-Polya operator
There is an old idea, usually attributed to Hilbert [1] and Polya [4] that the nontrivial(i.e. complex) zeros s n of the Riemann zeta function ζ ( s ) have a spectral interpretation.Writing s n := − i τ n , the Riemann hypothesis states that the nonimaginary solutions τ n of ζ ( − i τ n ) = 0 are real, that is the nontrivial zeros s n lie on the critical line Re s = . The Hilbert-Polya approach towards a proof of the Riemann hypothesisconsists in finding a Hilbert space H and a self-adjoint operator H in H whose discretespectrum is exactly given by the nontrivial zeros τ n = i (cid:0) s n − (cid:1) .Around 1950, Selberg [5] introduced his zeta function Z ( s ) in analogy with ζ ( s ) andwith the intention to shed some light on the nontrivial Riemann zeros and the Riemannhypothesis. He noticed the striking similarity between his famous trace formula forthe Laplace-Beltrami operator on e.g. compact Riemannian manifolds and the explicitformulae of number theory, whose most general form is Weil’s explicit formula [3].The nontrivial zeros of the Selberg zeta function Z ( s ) fulfil the analogue of Riemann’shypothesis and appear in the spectral side of the trace formula being directly relatedto the spectrum of the Laplacian. The other side of the trace formula has a purelygeometrical interpretation, since it is given by a sum over the length spectrum of theclosed geodesics (periodic orbits) of the geodesic flow, i.e. the free motion of a pointparticle on a given hyperbolic manifold. This system was already studied by Hadamard[6, 7] in 1898 and has played an important role in the development of ergodic theoryever since. Hadamard proved that all trajectories in this system are unstable andthat neighbouring trajectories diverge in time at an exponential rate, the most strikingproperty of deterministic chaos.In 1980, Gutzwiller [8] drew attention to this system as a prototype exampleof quantum chaos by identifying the Laplacian on hyperbolic manifolds with theSchrödinger operator in quantum mechanics. In this way he related the nontrivialzeros of the Selberg zeta function to the quantum energies of a dynamical system whoseclassical trajectories are chaotic. Furthermore, he realized that the Selberg trace formulais an exact version of his trace formula, the celebrated Gutzwiller trace formula [9], whichholds for general quantum systems with a chaotic classical counterpart, but in this caseonly approximately, i.e. in the so-called semiclassical limit where Planck’s constant ~ approaches zero.In 1985, Berry [10] emphasized that the search for the hypothetical Hilbert-Polyaoperator in terms of a Schrödinger operator obtained from the quantization of aclassically chaotic system might be a fruitful route to proving the Riemann hypothesis.He discussed in detail the properties of this operator that are suggested by the quantumanalogy. Prompted by a paper written by Connes [11] (see also [12]), who deviseda self-adjoint operator (Perron-Frobenius) of a classical dynamical system togetherwith a classical (Lefschetz) trace formula in noncommutative geometry, Berry andKeating [13, 14] speculated that the conjectured Hilbert-Polya operator might be somequantization of the extraordinarily simple classical Hamiltonian function H cl ( x, p ) of a he Berry-Keating operator on L ( R > , d x ) and on compact graphs x and its conjugate momentum p : H cl ( x, p ) := xp. (1)Inspired by [15, 16], Berry and Keating [13] suggested to investigate quantum graphmodels of (1), in particular the spectrum of these operators. One of the first researcherswho dealt with differential operators on graphs was Roth [17] who derived a trace formulafor the heat kernel of the Laplacian with Kirchhoff boundary conditions. Von Below[18] considered the heat equation on graphs and derived a characteristic equation for theeigenvalues of the weighted Laplacian on graphs. Some physical quantum graph modelswere considered by Exner and Šeba [19] who discussed i.a. the scattering problem for afree quantum particle on a star graph. A method to approximate mesoscopic systemslike thin branching systems by quantum graphs was discussed by Exner and Post [20]and Post [21]. Carlson [22] used semigroups on graphs to simulate the blood flow in thehuman arterial system. Kottos ans Smilansky [15, 16] introduced quantum graphs as amodel for quantum chaos.In this paper, we study the quantization of the classical Berry-Keating Hamiltonian(1) in the Hilbert space L ( R > , d x ) and on compact quantum graphs and give a completeclassification of the self-adjoint realizations of the corresponding Berry-Keating operator.In addition, we also study the quantization of the corresponding “squared” operator. Itturns out that no self-adjoint realization of (1) exists which yields as eigenvalues theRiemann zeros.
2. Classical dynamics and quantization of the Berry-Keating operator
Let us consider the classical dynamics of a particle moving on the real line R generated bythe Berry-Keating Hamiltonian (1) with corresponding phase space P : ( x, p ) ∈ R × R .The classical time evolution (Hamiltonian flow) is governed by Hamilton’s equations ˙ x ( t ) = ∂H cl ∂p = x ( t ) and ˙ p ( t ) = − ∂H cl ∂x = − p ( t ) . (2)Starting at time t = 0 at an arbitrary point ( x , p ) ∈ P in phase space, the uniquesolutions are [13] x ( t ) = x e t and p ( t ) = p e − t . (3)Obviously, the point (0 , ∈ P is an unstable point. We note that the Hamiltonian(1) is time independent corresponding to the conserved “energy” E := H cl ( x ( t ) , p ( t )) = x p ∈ R , and thus the particle moves in P on the “energy surface” (hyperbola) xp = E .Obviously, the classical motion is unbounded. Therefore, Berry and Keating [13, 14]introduced some regularization procedures, leading to a truncation of phase space, whichwe shall discuss below, but first we would like to discuss quantum mechanics.Quantization of the classical system requires to choose a Hilbert space H and toreplace the classical Hamiltonian (1) by a self-adjoint operator H in H . With thestandard choice H := L ( R , d x ) , the simplest operator corresponding to (1) is obtained he Berry-Keating operator on L ( R > , d x ) and on compact graphs x (acting by multiplication) and themomentum operator p = − i ~ dd x (acting by differentiation) leading to the Berry-Keatingoperator [13, 14] H BK := 12 ( xp + px ) = − i ~ (cid:18) x dd x + 12 (cid:19) , (4)and the Schrödinger equation i ~ ∂ Ψ( x, t ) ∂t = H BK Ψ( x, t ) . (5)As was to be expected from our discussion of the classical motion, the operator H BK isunbounded and does not have a discrete spectrum corresponding to bound states, butrather has a continuous spectrum λ ∈ R corresponding to scattering states obtained bysolving the eigenvalue problem H BK ψ ( x ) = λψ ( x ) . (6)Writing λ = ~ k , k ∈ R , Planck’s constant drops out from (6), and the eigenvalueproblem reads ( s := − + i k ) x d φ s ( x )d x = sφ s ( x ) . (7)For x ∈ R , (7) possesses the general solution φ s ( x ) = c x s + + c x s − , (8)where x s ± denote the generalized functions (see e.g. [23, p. 87]) x s + := ( for x ≤ x s for x > and x s − := ( | x | s for x < for x ≥ , (9)which is well defined for Re s > − . In [13], Berry and Keating studied as a special casethe simplest choice for the continuation of the eigenfunctions across the singularity at x = 0 by considering the even eigenfunctions ( c = c = c ) φ even s ( x ) = c | x | s .Let us discuss in more detail the case that the quantum dynamics takes place on thepositive half-line x ∈ R > . Then H BK acting on D ( R > ) , the set of infinitely continuousdifferentiable functions with compact support on R > , is essentially self-adjoint (see e.g.[24] [both deficiency indices are equal to zero]). Therefore, the closure of this operatoris self-adjoint. The general solution of the time-independent Schrödinger equation (6)is then given by ψ k ( x ) := 1 √ π x − +i k + with k ∈ R , (10)which is obviously not in L ( R > , d x ) and satisfies the orthonormality relation (in adistributional sense) h ψ k | ψ k ′ i := ∞ Z ψ k ( x ) ψ k ′ ( x )d x = δ ( k − k ′ ) (11) he Berry-Keating operator on L ( R > , d x ) and on compact graphs ∞ Z −∞ ψ k ( x ) ψ k ( x ′ )d k = δ ( x − x ′ ) . (12)Thus, we have for any φ ∈ L ( R > , d x ) the spectral decomposition φ ( x ) = ∞ Z −∞ A ( k ) ψ k ( x )d k (13)with A ( k ) := h ψ k | φ i = ∞ Z ψ k ( x ) φ ( x )d x (14)and (assuming h φ | φ i = 1 ) ∞ Z −∞ | A ( k ) | d k = 1 . (15)Forming a general wave packet with a given amplitude A ( k ) satisfying (13) and (15),one obtains ( x ∈ R > ) φ ( x ) = r πx ˆ A (ln x ) , (16)where ˆ A denotes the Fourier transform of A ˆ A ( y ) := 12 π ∞ Z −∞ A ( k )e i ky d k. (17)Let us also mention an alternative to the spectral decomposition (13) for the Berry-Keating operator. Defining the Mellin transform ˇ φ of φ by ˇ φ ( s ) := ∞ Z x s − φ ( x )d x, (18)we obtain for the wave number amplitude A ( k ) (see (14)) A ( k ) = 1 √ π ˇ φ (cid:16) − i k (cid:17) , (19)from which φ ( x ) can be recovered by the inverse Mellin transform (convergent at leastin mean square, see e.g. [25, p. 94]) φ ( x ) = 12 π i +i ∞ Z − i ∞ ˇ φ ( s ) x − s d s = 12 π ∞ Z −∞ ˇ φ (cid:16) − i k (cid:17) x − +i k d k = ∞ Z −∞ A ( k ) ψ k ( x )d k (20) he Berry-Keating operator on L ( R > , d x ) and on compact graphs U ( t ) := exp (cid:16) − i t ~ H BK (cid:17) = e − t e − tD (21)generated by the Berry-Keating operator (4) acts on functions φ ∈ H as (see i.a [26,p. 365]) ( U ( t ) φ )( x ) = e − t φ (cid:0) e − t x (cid:1) . (22)Here we have used the relation H BK = − i ~ (cid:0) D + (cid:1) , where D := x dd x is the generatorof scaling transformations (dilations). Let us mention that the operator D has beendiscussed by Arendt [27, 28, 29], where A p := − D is considered as the generator ofa semigroup on e.g. L p ( R > , d x ) ( ≤ p < ∞ ) with Dirichlet and Neumann boundaryconditions.On the other hand, the action of the unitary operator U ( t ) on eigenfunctions ψ of H BK gives according to (6) ( U ( t ) ψ )( x ) = e − i λ ~ t ψ ( x ) , (23)which in turn leads with (22), λ = ~ k , s = − + i k and κ := e − t > ( t < ∞ ) to ψ ( κx ) = κ s ψ ( x ) . (24)This shows that an eigenfunction ψ of H BK must be a homogeneous function with(complex) degree s = − + i k . Differentiation of (24) with respect to κ and then setting κ = 1 leads back to the eigenvalue problem (7) which possesses for x ∈ R > the uniquesolution (10)For the (retarded) integral kernel K BK ( x, x ; t ) of the time-evolution operator U ( t ) one obtains ( x, x ∈ R > ; Θ( t ) is the Heaviside step function) K BK ( x, x ; t ) = e − t δ (cid:0) x − x e − t (cid:1) Θ( t ) . (25)We observe that the quantum mechanical time evolution follows in the configurationspace exactly the classical trajectory (3). Starting at time t = 0 with the initial wavefunction φ ∈ L ( R > , d x ) , one obtains with (25) the wave function ψ ( x, t ) at a later time t > ψ ( x, t ) = ∞ Z K BK ( x, x ; t ) φ ( x )d x = e − t φ (cid:0) e − t x (cid:1) (26)in complete agreement with (22). We also give the result for the resolvent kernel(outgoing Green’s function [a small positive imaginary part ( ǫ > ) has been added he Berry-Keating operator on L ( R > , d x ) and on compact graphs λ = ~ k ]), see e.g. [30, p. 26], G BK ( x, x ; λ ) := i ~ ∞ Z e i ~ ( λ +i ǫ ) t K BK ( x, x ; t )d t = ∞ Z −∞ ψ k ′ ( x ) ψ k ′ ( x ) ~ k ′ − λ − i ǫ d k ′ = 2 π i ~ ψ k ( x ) ψ k ( x )Θ( x − x ) , (27)which satisfies the inhomogeneous time-independent Schrödinger equation (see (6)) ( H BK,x − ~ k ) G BK ( x, x ; ~ k ) = δ ( x − x ) . (28)Since the operator (4) acting in the Hilbert space L ( R , d x ) respectively L ( R > , d x ) has only a continuous spectrum, it cannot be considered (with the above realization) asa candidate for the hypothetical Hilbert-Polya operator. Thus, there remains the taskto find another Hilbert space for which the quantization of the classical Hamiltonian (1)possesses a discrete spectrum. Perhaps the required space is a quantum graph, with xp acting on edges between vertices, a possibility already mentioned by Berry and Keating[13]. It is the purpose of our paper to discuss the self-adjoint realizations on compactquantum graphs and in a forthcoming paper [31] we study a noncompact quantumgraph.
3. Semiclassical regularization of the Berry-Keating operator
Before we come to an investigation of quantum graphs, we would like to discuss analternative and very interesting approach also put forward by Berry and Keating [13](see also Connes [11, 12]) which is based on semiclassical arguments. It is well knownthat the number of quantum levels with energy less than E , the counting function N ( E ) ,is for any classical bounded Hamiltonian H cl ( x, p ) in one dimension given by (see e.g.[32])) N ( E ) = 12 π ~ area( E )(1 + O( ~ )) , (29)where area( E ) := Z P d x d p Θ( E − H cl ( x, p )) (30)is the phase-space area under the contour H cl ( x, p ) = E . Obviously, there is a problem ifthis formula is applied to the Hamiltonian (1), since the classical motion is not bounded,so that area( E ) is infinite. Therefore, Berry and Keating [13] proposed to regularizethe system by a suitable truncation of phase space in such a way that area( E ) becomesfinite.The regularization proposed by Berry and Keating [13] is to truncate x and p by considering the “regularized phase space” P reg := ( l x , ∞ ) × ( l p , ∞ ) together with he Berry-Keating operator on L ( R > , d x ) and on compact graphs l x l p = 2 π ~ . This truncation cuts off not only the “small”coordinate x ≤ l x respectively momentum values p ≤ l p , but it leads for a given “energy” E > also to a cut off at the “large” values x = El p respectively p = El x since E = H cl ( x, p ) holds. Without specifying the behaviour of the classical motion at the end points of thetrajectories, we follow Berry and Keating and obtain from (29) and (30) N ( E ) = 12 π ~ Elp Z l x Ex d x − l p (cid:18) El p − l x (cid:19) (1 + O( ~ ))= 12 π ~ E (cid:18) ln (cid:18) E π ~ (cid:19) − (cid:19) + 1 + . . . . (31)Setting ~ = 1 together with a modification of N ( E ) by adding − to the right-hand sideof (31) which was suggested by Berry and Keating [13, 14] in order to take into accountthe Maslov index, we arrive at the leading asymptotics of the counting function of thenontrivial zeros of the Riemann zeta function (Riemann- von Mangoldt formula) N ( E ) = E π ln (cid:18) E π (cid:19) − E π + 78 + O (ln E ) . (32)Following the argumentation of Berry and Keating [13, 14] for the modification of N ( E ) ,we get for the corresponding Maslov index µ = − . This seems at first somewhat strangesince there is no magnetic flux or spinning particle given and, therefore, the Maslovindex should be an integer number as in the case of “normal” quantum systems like theharmonic oscillator. We want to mention that there is actually no rigorous argument forthe choice of the Maslov index (correction) simply by the fact that so far we have not yetimposed any boundary conditions on the operator, and in the corresponding classicaldescription there is therefore a lack of jump or scattering condition at the end points ofthe trajectories. The scattering conditions in section 16 (example 16.2) could provide apossible remedy for the above mentioned discrepancy of the Maslov index with respectto “normal” systems. Furthermore, there is only one possibility in the classical case forthe behaviour of the particle at the end point of the trajectory if one wants to preservethe constancy of the Hamiltonian for all time: the particle must jump from the point ( El p , l p ) to the point ( l x , El x ) in phase space, which corresponds to a kind of ring-system(one-dimensional torus with the topology of S ) in the configuration space.
4. Classical dynamics and quantization of the “squared” Berry-Keatingoperator
In order to allow some kind of reflection at the end points of the trajectories, we shallalso consider the classical Hamiltonian e H cl ( x, p ) := x p , (33) he Berry-Keating operator on L ( R > , d x ) and on compact graphs L ( x, ˙ x ) = 14 (cid:18) ˙ xx (cid:19) (34)and that Hamilton’s equations do not decouple in this case as in (2). In fact, one obtains ˙ x ( t ) = ∂ e H cl ∂p = 2 x p ( t ) and ˙ p ( t ) = − ∂ e H cl ∂x = − xp ( t ) , (35)and the solutions are x ( t ) = x e x p t and p ( t ) = p e − x p t . (36)If one broadens the phase space to P reg ,b := ( l x , ∞ ) × (( l p , ∞ ) ∪ ( − l p , −∞ )) (37)one now has the possibility to scatter from the end point ( El p , l p ) of a trajectory of theform (36) to the end point ( El p , − l p ) . This corresponds to a reflection on a wall like ina one-dimensional billiard system. This is one reason why we rather consider H cl andaccordingly H BK as a momentum (operator) and e H cl and respectively H as an energy(operator). Further hints to this choice will follow in the sequel.Before investigating the “squared” Berry-Keating operator on quantum graphs, wewould like to consider this operator in the framework of standard quantum mechanicsrestricting ourselves, however, to the positive half-line R > as in the discussion of theoriginal Berry-Keating operator in section 2. A formal calculation of e H := H gives(setting from now on ~ = 1 ): H := (cid:18) − i (cid:16) x dd x + 12 (cid:17)(cid:19) = − x d d x − x dd x − . (38)Again as in section 2, H acting on D ( R > ) is essentially self-adjoint, and in thefollowing we always consider the self-adjoint closure of this operator. It is worthwhileto mention that the squared operator (38) is a special case of the famous Black-Scholesoperator [33, 34] introduced to determine the pricing of options in financial theory whoseinteresting mathematical properties have been discussed e.g. in [27, 28, 29].It is easy to see that the functions ψ k ( x ) ( k ∈ R \ { } ) defined in (10) are the onlyeigenfunctions of H on R > corresponding to the continuous spectrum λ = k > .Here the eigenvalue λ = 0 (respectively k = 0 ) corresponds to the two generalizedeigenfunctions ψ , ( x ) = 1 √ π x − + and ψ , ( x ) = 1 √ π x − + ln x. (39)An eigenvalue λ = k > possesses the two linearly independent generalizedeigenfunctions ψ k ( x ) and ψ − k ( x ) .Introducing the (retarded) integral kernel of the time-evolution operator (unitarygroup) e U ( t ) := e − i tH (40) he Berry-Keating operator on L ( R > , d x ) and on compact graphs ψ ( x, t ) := (cid:16) e U ( t ) φ (cid:17) ( x ) =: Z R > e K ( x, x ; t ) φ ( x )d x , (41)where φ ( x ) ∈ L ( R > , d x ) is the initial wave function at t = 0 , we obtain (cf. [30, p. 27]) e K ( x, x ; t ) = ∞ Z −∞ ψ k ( x ) ψ k ( x )e − i k t Θ( t )d k = (4 π i txx ) − e i (ln x − ln x t Θ( t ) . (42)The kernel e K satisfies the inhomogeneous time-dependent Schrödinger equation (cid:18) i ∂∂t − H BK,x (cid:19) e K ( x, x ; t ) = i δ ( x − x ) δ ( t ) , (43)i.e. it is the retarded Green’s function. With (41), the action of e U ( t ) on φ ∈ L ( R > , d x ) is given by ( t > ) (cid:16) e U ( t ) φ (cid:17) ( x ) = (4 π i t ) − ∞ Z −∞ e i τ t e − τ φ (cid:0) e − τ x (cid:1) d τ, (44)which expresses the fact that e U ( t ) is a combination of the scaling transformationgenerated by the operator D = x dd x (see eq. (22)) and the transformation generatedby the operator T := x d x , since e U ( t ) = e i t e i tT e tD . Notice that the transformationgenerated by T reads (cid:0) e i tT φ (cid:1) ( x ) = e − i t (4 π i t ) ∞ Z −∞ e i τ t + τ φ (cid:0) e − τ x (cid:1) d τ, (45)and that the operators D and T commute. The resolvent kernel (outgoing Green’sfunction) of H is given by (see [30, p. 26]) e G ( x, x ; λ ) := i ∞ Z e i( λ +i ǫ ) t e K ( x, x ; t ) d t = (4 xx ( − λ − i ǫ )) − e − ( − λ − i ǫ ) | ln x − ln x | , (46)which shows that e G has a cut on the positive real axis in the complex λ -plane (if √ z is defined with a cut on the negative real axis in the z -plane). With k := √ λ > oneobtains ( x, x ∈ R > ) e G (cid:0) x, x ; k (cid:1) = i2 k √ xx e i k | ln x − ln x | = i πk ( ψ k ( x ) ψ k ( x ) for x ≥ x ψ k ( x ) ψ k ( x ) for x < x (47)in agreement with the general form of the Green’s function of a Sturm-Liouville operator(see e.g. [35, p. 112]). he Berry-Keating operator on L ( R > , d x ) and on compact graphs
5. Semiclassical estimate for the eigenvalue counting function of the“squared” Berry-Keating operator
Using again the semiclassical formula (29) and the truncation of phase space as discussedin section 3, we obtain for the counting function in the quadratic case N ( E ) = 12 π ~ √ Elp Z l x √ Ex d x − l p √ El p − l x ! (1 + O( ~ ))= 2 (cid:20) k π ln (cid:18) k π (cid:19) − k π + 78 (cid:21) + . . . , (48)where we have included the same Maslov index correction as in section 3. Furthermore,we have introduced the “wave number” k , E =: ~ k , and have used l x l p = 2 π ~ . Wenote that in this case we obtain twice the counting function of the Riemann zeros (forwhich only those with positive imaginary part are counted), since each energy value E comes with two values ± k . Notice, that in this case the Riemann zeros are notinterpreted as “energies” but rather as “momenta” ~ k respectively “wave numbers” k .Formula (48) agrees with the well-known universal law that N ( E ) for a bounded systemin d dimensions grows asymptotically as N ( E ) = O( E d ) , and thus for a one-dimensionalsystem one expects N ( E ) = O( √ E ) = O( k ) , eventually modified by a factor ln( √ E ) .
6. Compact graphs
We shall present a short overview on compact graphs using the notations of [36] and[37]. A compact graph
Γ = ( V , E , I ) is a finite set of vertices V = ( v , . . . , v V ) and a finiteset of edges E = ( e , . . . , e E ) . Here we have defined E := |E | and V := |V| for the totalnumber of edges and vertices, respectively. Each vertex v ∈ V is at least connected withone element ˜ v ∈ V by some edge e ∈ E , where v = ˜ v is allowed. Furthermore, each edge e ∈ E connects two vertices v and ˜ v in V , again v = ˜ v is possible. The topology of thegraph is given by these relations of the edges and the vertices. Each edge e is assignedan interval I e = [ a e , b e ] with < a e < b e < ∞ . The set of all intervals is denoted by I .We remark that the choice of the starting point a e and the final point b e of the edge e is arbitrary and there is no orientation of the graph assumed. We denote two edges asadjacent iff they share at least one vertex as endpoint. We need the notion of a path andof a periodic orbit of the graph. We slightly differ from the definition in [37] for furtherconvenience. A path p ( w, z ) := (( e i ) ni =1 , w, z ) is a set of a finite sequence of edges ( e i ) ni =1 where the points w, z ∈ I denote the starting and final points of the path. Furthermore,it is required that • the edges e i and e i +1 are adjacent, • the point w must be an element of I and z must be an element of I n . he Berry-Keating operator on L ( R > , d x ) and on compact graphs w = z is admissible and corresponds to a closed path. In [37] or [16] only thefirst item is required for a closed path at which we set e = e n +1 . We shall call this casea closed orbit. Especially, a closed orbit is only characterized by a sequence of edges ( e i ) ni =1 . For the definition of a periodic orbit γ , we shall keep with the usual definitionas in [37], then γ is an equivalence class of closed orbits and can be characterized bya representative γ = ( e i ) ni =1 . The set of all periodic orbits is denoted by P . We couldthen equip the graph with a metric structure in an obvious way like in [37]. Especially,this would mean that the length of the edge e i will be l i = b i − a i . However, here wetake another choice for the lengths and the metric structure of the graph. We define thelength l p ( w, z ) of the path p ( w, z ) := (( e i ) ni =1 , w, z ) as follows. If n ≥ we denote by y and y n the endpoints of the intervals I and I n corresponding to the shared vertices ofthe edges e , e and e n − , e n . In particular this means that y is identical with a or b and y n is identical with a n or b n . Then the length l p ( w, z ) is defined as l p ( w, z ) := (cid:12)(cid:12)(cid:12) ln (cid:16) y w (cid:17)(cid:12)(cid:12)(cid:12) + n − X i =2 ln (cid:18) b i a i (cid:19) + (cid:12)(cid:12)(cid:12) ln (cid:16) y n z (cid:17)(cid:12)(cid:12)(cid:12) . (49)Similarly, if n = 2 ( y = y n := y ) respectively n = 1 we define l p ( w, z ) := (cid:12)(cid:12)(cid:12) ln (cid:16) yw (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ln (cid:16) yz (cid:17)(cid:12)(cid:12)(cid:12) respectively l p ( w, z ) := (cid:12)(cid:12)(cid:12) ln (cid:16) wz (cid:17)(cid:12)(cid:12)(cid:12) . (50)Furthermore, we define in a natural way the length l γ of a periodic orbit γ l γ := ln n Y i =1 b i a i ! . (51)In order to define a metric structure of the graph, we need the notion of connectedness.We define w and z as connected iff there exists a path p ( w, z ) := (( e i ) ni =1 , w, z ) . Thegraph Γ is connected iff all points of the intervals I are connected. Not necessarily butfor convenience, we assume in the following that the graph Γ is connected. The distance d w,z of two points w and z on the edges of the graph is defined by d w,z := min { l p ; p connects w and z } . (52)We remark that this choice of the metric for the graph will correspond to a “hyperbolic”metric in one dimension. In this case the determinant of the metric tensor g at thepoint x is (det g )( x ) = x . The reason for our choice of the metric will be explained insections 7 and 13.We also need for the interpretation of the trace formula for the Berry-Keatingoperator H BK in theorem 15.2 the notion of a directed graph in order to interpret theright side of (136) as a sum of periodic orbits. Therefore, we replace the edges by directededges. This doesn’t affect the lengths of the edges but has of course an influence ontopological properties of the graph such as connectedness and on the set P of periodicorbits. Since in this case there are only such paths p ( w, z ) := (( e i ) ni =1 , w, z ) allowed forwhich for all consecutive edges e i , e i +1 , there exist vertices v ij such that the direction of e i is towards v ij and e i +1 has the direction away from v ij . Then, the definition for theperiodic orbits and for connectedness for directed graphs are the same as for undirected he Berry-Keating operator on L ( R > , d x ) and on compact graphs H BK ) is connected.
7. The Berry-Keating operator on compact quantum graphs
We define, in accordance with [37]: C ∞ (Γ) := E M i =1 C ∞ [ a i , b i ] and H := L (Γ) := E M i =1 L ([ a i , b i ] , d x ) with < a i < b i < ∞ . (53)This means that a function of the Hilbert space H is represented by an orthogonal sumof functions which are defined on the corresponding edges: ψ ∈ L (Γ) iff ψ = E M i =1 ψ i with ψ i ∈ L ( I i , d x ) . (54)The first function space in (53) will be a possible operator core for the closedBerry-Keating operator with (57) as domain of definition. Therefore, the self-adjointextensions of H BK are also with respect to C ∞ (Γ) . The space H is a Hilbert space if weequip it with the scalar product h ψ, φ i := E X j =1 b j Z a j ψ j ( x j ) φ j ( x j )d x j . (55)We then define the Berry-Keating operator on compact graphs (in the following we set ~ = 1 ): H BK ψ := (cid:18) − i (cid:16) x dd x + 12 (cid:17) ψ , . . . , − i (cid:16) x dd x + 12 (cid:17) ψ E (cid:19) , (56)for ψ ∈ C ∞ (Γ) . Since we have a compact graph Γ , multiplication by x is a boundedclosable operation. Thus, by perturbation arguments, see e.g. [38, p. 183], we concludethat H BK is closable since the standard momentum operator p = − i dd x is closable.Furthermore, we note that multiplication by the argument is also a (bounded) bijectionfrom D (Γ) := E M i =1 H [ a i , b i ] (57)to itself. H [ a i , b i ] is the set of absolutely continuous functions on [ a i , b i ] which vanishat the endpoints of the intervals and with square integrable weak derivatives. Again,by perturbation arguments for the momentum operator, we therefore conclude that thedomain of definition of the closure of H BK is equal to (57). Furthermore, by similararguments the adjoint operator of ( H BK , D (Γ)) is given by ( H BK , H (Γ)) in which H (Γ) := E M i =1 H [ a i , b i ] (58) he Berry-Keating operator on L ( R > , d x ) and on compact graphs Γ , cf. [38,p. 100] possessing square integrable weak derivatives.We mention at this point that the projections of the spaces D (Γ) and H (Γ) onthe intervals of the graph Γ coincide with the corresponding Sobolev spaces, see e.g. [39].The operator ( H BK , D (Γ)) is symmetric and it is possible to show that the deficiencyindices are ( E, E ) , compare e.g. [24, p. 142]. By a proper Sobolev embedding theoremand the compactness of the graph Γ it follows that the differential operator on D (Γ) possesses a compact resolvent, see also [40]. Thus, by the compact resolvent theorem,cf. [41, p. 245], and the relatively compact perturbation theorem (cf. [41, p. 113]) theoperator (56) possesses a purely discrete spectrum.
8. Classification of the self-adjoint extensions of the Berry-Keating operator
In order to characterize the self-adjoint extensions, we follow the ideas of [24, p. 138]and [36], see also [42] for a comprehensive discussion. Therefore, we define the complexsymplectic form on H (Γ) × H (Γ) (cf. [24, p. 138]): [ φ, ψ ] := (cid:10) φ, H +BK ψ (cid:11) L (Γ) − (cid:10) H +BK φ, ψ (cid:11) L (Γ) for φ, ψ ∈ H (Γ) . (59)We call a subspace X [ · , · ] -symmetric, iff [ φ, ψ ] = 0 for all φ, ψ ∈ X . Due to thevon Neumann extension theory (see e.g. [24]) the self-adjoint extensions are exactly themaximal [ · , · ] -symmetric subspaces of H (Γ) . We follow the approach by Kostrykin andSchrader [36] to classify these extensions. By a proper Sobolev embedding theorem wecan define the boundary value bv (an element of C E ) of ψ ∈ H (Γ) : Ψ bv := ( ψ ( a ) , . . . , ψ E ( a E ) , ψ ( b ) , . . . , ψ E ( b E )) T for ψ ∈ H (Γ) . (60)For convenience, we also define: I ± := E × E − E × E ! , D ( ab ) := a b ! , (61)with (no summation over i ) a ij := δ ij a i and b ij := δ ij b i for ≤ i, j ≤ E (62)and J := E × E − E × E ! , U := 1 √ i E × E E × E − E × E − i E × E ! . (63)By a simple calculation we obtain the identity: U (i I ± ) U + = J. (64)Thus, we obtain for (59) by an integration by parts using the unitarity of U : [ ψ, φ ] = (cid:10) Φ bv , i I ± D ( ab ) Ψ bv (cid:11) C E = D D ( ab ) Φ bv , i I ± D ( ab ) Ψ bv E C E = D U D ( ab ) Φ bv , J U D ( ab ) Ψ bv E C E for all ψ, φ ∈ H (Γ) . (65) he Berry-Keating operator on L ( R > , d x ) and on compact graphs D ( ab ) we have used the usual definition of a positive operator([39, p. 196]), which in this case simply means to take the square root of the (diagonal)entries in D ( ab ) . Note that ω ( · , · ) := h· , J ·i C E (66)defines a nondegenerate complex symplectic form on C E × C E . We call a subspace L of C E a Lagrangian subspace iff • a, b ∈ L then ω ( a, b ) = 0 . • Whenever for a subspace ˜ L ⊃ L the first property holds,it follows ˜ L = L . (67)For the Lagrangian subspaces of C E we apply the result of [36]. A subspace L isLagrangian iff there exist two matrices A, B ∈ Mat( E × E, C ) with: AB + = BA + and rank( A, B ) = E. (68)We then have, L = ( φ ∈ C E ; φ := φ φ ! and Aφ + Bφ = 0 ) . (69)In (68) the matrix ( A, B ) is formed of the columns of A and B , and we have introducedtwo maps ( · ) i : C E → C E for ≤ i ≤ (70)by φ i := ( ( , φ if i = 1 , (0 , ) φ if i = 2 . (71)Furthermore, as mentioned in [43], these matrices are not uniquely defined. Two sets ofmatrices A, B and e A, e B define the same Lagrangian subspace iff there exists an invertiblematrix C with A = C e A and B = C e B. (72)Since U D ( ab ) is a bijection from C E onto itself and by (59) and (65) we infer, withthe same arguments as in [36], that there is a one-to-one correspondence between theself-adjoint extensions of ( H BK , D (Γ)) and the Lagrangian subspaces of C E . We thushave proved the following proposition. Proposition 8.1
Each domain of definition of such a self-adjoint extension is exactlythe preimage with respect to (60) of a subspace L = n Ψ bv ∈ C E ; A (cid:16) U D ( ab ) Ψ bv (cid:17) + B (cid:16) U D ( ab ) Ψ bv (cid:17) = 0 o , (73) where A and B fulfil (68). The converse is also true. Because of proposition 8.1, we denote in the following the self-adjoint extensions of ( H BK , D (Γ)) by ( H BK ; A, B ) . he Berry-Keating operator on L ( R > , d x ) and on compact graphs
9. Determination of the eigenvalues of H BK All possible eigenfunctions ψ k to an eigenvalue k of H BK are of the form ψ k ( x ) = (cid:18) α √ x e i k ln x , . . . , α E √ x e i k ln x (cid:19) . (74)We denote the column vector (60) corresponding to ψ k by Ψ bv,k . In order to apply (73)for determining the eigenvalues k and the corresponding eigenvectors ψ k , we calculate U D ( ab ) Ψ bv,k using (63). U D ( ab ) Ψ bv,k = U e i k ln a
00 e i k ln b ! α = 1 √ ie i k ln a e i k ln b − e i k ln a − ie i k ln b ! α = 1 √ ie i k ln a + e i k ln b − e i k ln a − ie i k ln b ! α . (75)For convenience, we have used the notations: (cid:0) e i k ln a (cid:1) mn := δ mn e i k ln a m , (cid:0) e i k ln b (cid:1) mn := δ mn e i k ln b m for ≤ m, n ≤ E (76)and α := ( α , . . . , α E , α , . . . , α E ) T . (77)Therefore, we get: √ (cid:16) U D ( ab ) Ψ bv,k (cid:17) = (cid:0) ie i k ln a + e i k ln b (cid:1) α √ (cid:16) U D ( ab ) Ψ bv,k (cid:17) = − (cid:0) e i k ln a + ie i k ln b (cid:1) α , (78)where we have used ( · ) i defined in (71).Taking into account that α = α =: ˜ α , we obtain for the expression in (73) √ h A (cid:16) U D ( ab ) Ψ bv,k (cid:17) + B (cid:16) U D ( ab ) Ψ bv,k (cid:17) i = (cid:16) A (cid:0) ie i k ln a + e i k ln b (cid:1) − B (cid:0) e i k ln a + ie i k ln b (cid:1)(cid:17) ˜ α = (cid:0) i ( A + i B ) e i k ln a + ( A − i B ) e i k ln b (cid:1) ˜ α . (79)Kostrykin and Schrader have shown, see [36], that A ± i B are invertible under theassumption (68). With the notation C ( k ) := e i k ln a ˜ α (80)and, because ( A − i B ) and ( A + i B ) − commute (see [44]), we get: A (cid:16) U D ( ab ) Ψ bv,k (cid:17) + B (cid:16) U D ( ab ) Ψ bv,k (cid:17) = 0 ⇔ (cid:18) − i A − i BA + i B e i k ln ba (cid:19) C ( k ) = 0 , (81) he Berry-Keating operator on L ( R > , d x ) and on compact graphs e i k ln ba similarly defined as in (76). Due to the similarity of (81) with the secularequation for the common Laplace operator on compact graphs (see [43]), we denote: S ( A, B ) := i A − i BA + i B and T ( a , b ; k ) := e i k ln ba . (82)It follows with exactly the same arguments as in [36] that S ( A, B ) is unitary, and weshall call it also the S -matrix of the quantum graph. The unitarity of T ( a , b ; k ) iff k ∈ R is obvious. Since, for all k ∈ C , C ( k ) = 0 iff ˜ α = 0 , we have proved the followingproposition establishing the secular equation F ( k ) = 0 . Proposition 9.1 k n ∈ R is an eigenvalue of ( H BK ; A, B ) iff F ( k n ) := det( E × E − S ( A, B ) T ( a , b ; k n )) = 0 . (83) Furthermore, the multiplicity of the eigenvalue one of S ( A, B ) T ( a , b ; k n ) coincides withthe multiplicity of the eigenvalue k n of ( H BK ; A, B ) . We remark that the geometric multiplicities and the algebraic multiplicities of S ( A, B ) T ( a , b ; k ) coincide since this matrix is diagonalizable.In contrast to the S -matrix of the generic negative Laplacian − ∆ on graphs, theS-matrix S ( A, B ) is always independent of k (the S -matrix of − ∆ is independent ofthe wave number k iff S + = S [see [43]]). But we remark that the independence ofthe S -matrix on the eigenvalue will also occur when we replace H BK by the standardmomentum operator (with x ∈ R ) p := − i dd x . (84)The calculations are quite analogous (see e.g. [45] for a detailed discussion of this). Infact every self-adjoint extension of p can be characterized by the same matrices A and B as in (68) and we would get the same S -matrix S ( A, B ) . The only difference in thesecular equation between the operators H BK and p then is the form of the second matrixin (82) which in the case of p is given by T ( a , b ; k ) := e i k ( b − a ) . (85)This is one reason why we rather relate H BK with a momentum operator than an energyoperator as indicated in section 2. Because of the occurrence of the logarithm in T weendow the quantum graph with a metric structure as proposed in section 6. We alsomention that proposition 9.1 is valid for all k ∈ C , in particular for k = 0 in contrast tothe corresponding proposition 12.1 for H BK . However, the analogy between p = − ∆ and H is not so obvious.
10. The “squared” Berry-Keating operator
Our Hilbert space will be H = L (Γ) , see (54) and (55). Again, we seek self-adjointextensions of (38) with respect to C ∞ (Γ) . In order to obtain a self-adjoint operator, thetask is to specify an appropriate domain D ( H ) for this operator with C ∞ (Γ) ⊂ D ( H ) . (86) he Berry-Keating operator on L ( R > , d x ) and on compact graphs H as the “squared” Berry-Keating operator, whichmeans: H ψ := H BK ( H BK ψ ) ,ψ ∈ D ( H ) := { φ ∈ D ( H BK ); H BK φ ∈ D ( H BK ) } . (87)It follows immediately that H is self-adjoint if H BK is self-adjoint using Friedrichs’extension theorem [24, p. 180]. But in fact there are many possible self-adjoint extensionswhich cannot be realized in such a way. We will give simple examples in section 16. Wecan generalize these constructions to consider non-self-adjoint but closed realizations of H BK and then form H +BK H BK or H BK H +BK . (88)This is an idea quite analogous to the concept of supersymmetry, see [46] and [47] (thetechnique of factorization was already introduced by Schrödinger [48] and reviewed in[49]). In [47] this technique has been used but isn’t explicitly mentioned. However,only a certain kind of self-adjoint extension can be attained in such a way. In [47] theseare exactly the self-adjoint extensions which correspond to k -independent S -matricescorresponding to these extensions. This relation between the S -matrices and the self-adjoint extensions of the negative Laplace operator − ∆ on metric graphs is explainedin [43].We would like to give an overview of the starting point of our considerations froma mathematical point of view. The proofs of these statements are similar as in section7 using the same references as there. Therefore, we only summarize the results. First,the operator H acting on C ∞ (Γ) or D (Γ) := E M i =1 H [ a i , b i ] with < a i < b i < ∞ (89)is symmetric. H [ a i , b i ] is the set of absolutely continuous functions which possessabsolutely continuous derivatives on [ a i , b i ] , square integrable weak second derivativesand which together with their first derivatives vanish at the endpoints of the intervals.Furthermore, H acting on D (Γ) is closed and the adjoint operator of ( H , D (Γ)) is ( H , H (Γ)) . Here H (Γ) := E M i =1 H [ a i , b i ] with < a i < b i < ∞ (90)is the space of functions being absolutely continuous on the intervals of the graph Γ possessing absolutely continuous derivatives and weak square integrable secondderivatives. The deficiency indices are (2 E, E ) , thus ( H , D (Γ)) possesses infinitelymany self-adjoint extensions. Again, as for H BK the spectrum of every self-adjointextension is purely discrete and as in section 7 the projections of the spaces D (Γ) and H (Γ) on the intervals of the graph Γ coincide with the corresponding Sobolev spaces(see again e.g. [39]). We shall follow a general approach to find all these self-adjointextensions, quite analogous as in section 8 and based on [36]. he Berry-Keating operator on L ( R > , d x ) and on compact graphs
11. Classification of the self-adjoint extensions of the “squared”Berry-Keating operator
First, we define Ψ bv as in (60) for ψ ∈ H (Γ) and additionally Ψ ′ bv := ( ψ ′ ( a ) , . . . , ψ ′ E ( a E ) , − ψ ′ ( b ) , . . . , − ψ ′ E ( b E )) T for ψ ∈ H (Γ) , (91)in which ψ ′ i is the derivative of ψ i on the interval I i . Similarly as in (59), we define asymplectic form on H (Γ) × H (Γ)[ φ, ψ ] := D φ, H ψ E L (Γ) − D H φ, ψ E L (Γ) for φ, ψ ∈ H (Γ) . (92)With the same arguments as for H BK in section 7 the task is to find all maximal [ · , · ] -symmetric subspaces of H (Γ) in order to find all self-adjoint extensions to ( H , D (Γ)) .We shall adapt the definition of J in (63) by J := E × E − E × E ! (93)and (see (61)) e D ( ab ) := D ( ab ) D ( ab ) ! , [ ψ ] bv := Ψ bv Ψ ′ bv ! for ψ ∈ D (Γ) . (94)We obtain for φ, ψ ∈ H (Γ) using partial integration and the fact that J and e D ( ab ) commute: [ ψ, φ ] = E X i =1 (cid:16) b i (cid:16) ψ ′ i ( b i ) φ ( b i ) − ψ i ( b i ) φ ′ ( b i ) (cid:17) − a i (cid:16) ψ ′ i ( a i ) φ ( a i ) − ψ i ( a i ) φ ′ ( a i ) (cid:17)(cid:17) = D [ ψ ] bv , J e D ab ) [ φ ] bv E C E = D e D ( ab ) [ ψ ] bv , J e D ( ab ) [ φ ] bv E C E . (95)Taking the scalar product in the definition of ω ( · , · ) in (66) with respect to C E , we inferas in section 8 the following proposition. Proposition 11.1
The self-adjoint extensions of ( H , D (Γ)) are exactly thepreimages of L = n [ φ ] bv ∈ C E ; A (cid:16) e D ( ab ) [ φ ] bv (cid:17) + B (cid:16) e D ( ab ) [ φ ] bv (cid:17) = 0 o = (cid:8) [ φ ] bv ∈ C E ; AD ( ab ) Φ bv + BD ( ab ) Φ ′ bv = 0 (cid:9) (96) with respect to [ · ] bv in (94). In (96) we have used ( · ) i defined in (71). The matrices A and B are now elements of Mat(2 E × E, C ) with the adopted conditions AB + = BA + and rank( A, B ) = 2 E. (97)Again, as in section 8 and because of proposition 11.1 we denote the self-adjointextensions of ( H , D (Γ)) by ( H ; A, B ) . he Berry-Keating operator on L ( R > , d x ) and on compact graphs
12. Determination of the eigenvalues of H We want to solve the eigenvalue problem H ψ = λψ. (98)To tackle this problem, it will be convenient to consider the wave number k defined by λ := k . It is a trivial observation that ± k correspond to the same eigenvalue λ . Thisfact will be revealed in the symmetry of the secular equation for the wave number.In addition to (98) the eigenvector ψ must be in the domain of definition of theoperator. However, the general form of the eigenvector to an eigenvalue λ = k = 0 is ψ k ( x ) = (cid:18) √ x (cid:0) α e i k ln x + β e − i k ln x (cid:1) , . . . , √ x (cid:0) α E e i k ln x + β E e − i k ln x (cid:1)(cid:19) . (99)We can proceed as in [36]. Therefore, we compute Ψ bv,k and Ψ ′ bv,k using the definitionsin (61) and (76). Ψ bv,k = D − ( ab ) e i k ln a e − i k ln a e i k ln b e − i k ln b ! αβ ! Ψ ′ bv,k = − D − ( ab ) e i k ln a e − i k ln a − e i k ln b − e − i k ln b ! +i kD − ( ab ) e i k ln a e − i k ln a − e i k ln b − e − i k ln b ! I ± ! αβ ! . (100)In order to be in the domain of definition of a self-adjoint realization, Ψ bv,k and Ψ ′ bv,k mustbe in some L of (96) defined by the two matrices A and B . We make the identification(cf. [43]), X ( k ; a , b ) := e i k ln a e − i k ln a e i k ln b e − i k ln b ! , (101) Y ( k ; a , b ) := e i k ln a e − i k ln a − e i k ln b − e − i k ln b ! and (102) Y ′ ( k ; a , b ) := e i k ln a − e − i k ln a − e i k ln b e − i k ln b ! . (103)Thus, we conclude, with the definition for the bold symbols in accordance with (62): ! = AD ( ab ) Φ bv + BD ( ab ) Φ ′ bv (104) = AD ( ab ) X ( k ; a , b ) + BD − ( ab ) Y ( k ; a , b ) − + i k − − i k !! αβ ! = (cid:18)(cid:16) AD ( ab ) − BD − ( ab ) I ± (cid:17) X ( k ; a , b ) + i kBD − ( ab ) Y ′ ( k ; a , b ) (cid:19) αβ ! . (105) he Berry-Keating operator on L ( R > , d x ) and on compact graphs D − ( ab ) is self-adjoint, we conclude AD ( ab ) (cid:16) BD − ( ab ) (cid:17) + = AB + . (106)Since D − ( ab ) and D ( ab ) are invertible and diagonal, it is easy to show that rank( AD ( ab ) , BD − ( ab ) ) = rank( A, B ) = 2 E. (107)Therefore, we define AD ( ab ) =: A ′ and BD − ( ab ) =: B ′ (108)and observe that A ′ and B ′ also fulfil the conditions (97). Therefore, we can apply atheorem of Kuchment [40]. It states that two matrices A ′ and B ′ fulfil (97) iff thereexists an invertible matrix C with: A ′ = CP ker B ′ + CP ⊥ ker B ′ L ′ P ⊥ ker B ′ and B ′ = CP ⊥ ker B ′ . (109)In (109) we have defined P ker B ′ as the projector onto the kernel of B ′ and P ⊥ ker B ′ as thecorresponding orthogonal projector. The matrix L ′ is self-adjoint and can be defined by(see [40]), L ′ := ( B ′ | ran B ′ + ) − A ′ P ⊥ ker B ′ . (110)Hence, we can proceed in the calculation (104) by multiplying (105) from the left-handside by C − (cid:18)(cid:16) P ker B ′ + P ⊥ ker B ′ L ′ P ⊥ ker B ′ − P ⊥ ker B ′ I ± (cid:17) X ( k ; a , b )+ i kP ⊥ ker B ′ Y ′ ( k ; a , b ) (cid:19) αβ ! . (111)Since the projectors P ker B ′ and P ⊥ ker B ′ are mutually orthogonal, we infer from (109), thedefinition of L in (96) and with the first line of (105), that P ker B ′ X ( k ; a , b ) αβ ! = 0 . (112)In (111) we insert between the matrices I ± and X ( k ; a , b ) the unit matrix = P ker B ′ + P ⊥ ker B ′ and apply (112) (cid:18)(cid:16) P ker B ′ + P ⊥ ker B ′ (cid:16) L ′ − I ± (cid:17) P ⊥ ker B ′ (cid:17) X ( k ; a , b )+ i kP ⊥ ker B ′ Y ′ ( k ; a , b ) (cid:19) αβ ! . (113)We realize that L ′ − I ± is also self-adjoint. Thus, we define L ′′ := P ⊥ ker B ′ (cid:18) L ′ − I ± (cid:19) P ⊥ ker B ′ (114) he Berry-Keating operator on L ( R > , d x ) and on compact graphs A ′′ := P ker B ′ + L ′′ and B ′′ := P ⊥ ker B ′ . (115)It is obvious that the matrices A ′′ and B ′′ fulfil the conditions (95). Hence as in section9 respectively [36], we infer that A ′′ ± i kB ′′ is invertible and conclude with quite thesame calculation as in [43] A ′′ + i kB ′′ ) [ − S ′′ ( A, B ; k ) T ( a , b ; k )] e i k ln a
00 e − i k ln b ! αβ ! . (116)Here we have used the definitions S ′′ ( A, B ; k ) := S ( A ′′ , B ′′ ; k ) := − A ′′ − i kB ′′ A ′′ + i kB ′′ and T ( a , b ; k ) := i k ln ba e i k ln ba ! . (117)The first and the third matrix in the product of (116) are invertible for all k ∈ C \ ( ± i σ ( L ′′ )) in which σ ( L ′′ ) denotes the spectrum of L ′′ . For a detailed discussionof this, see [37] and [44]. Thus, we have proved the following proposition. Proposition 12.1 k with k ∈ C \ ( ± i σ ( L ′′ ) ∪ { } ) is an eigenvalue of ( H ; A, B ) iff F ( k ) := det ( E × E − S ′′ ( A, B ; k ) T ( a , b ; k )) = 0 . (118) Furthermore, as in section 9, the multiplicity of the eigenvalue λ = k coincideswith the multiplicity of the eigenvalue one of S ′′ ( A, B ; k ) T ( a , b ; k ) for every k ∈ C \ ( ± i σ ( L ′′ ) ∪ { } ) . We remark that the restriction on k concerns only non-positive eigenvalues λ = k of H .
13. The eigenvalue zero
For the eigenvalue λ = 0 , which is equivalent to the case k = 0 , the eigenfunctions areof the form ψ ( x ) = (cid:18) α √ x + β √ x ln x, . . . , α E √ x + β E √ x ln x (cid:19) . (119)With a similar calculation as for the case k = 0 one obtains the equation A ′′ E × E ln a E × E ln b ! + B ′′ E × E − E × E !! αβ ! = 0 (120)which is necessary and sufficient for λ = 0 to be an eigenvalue of ( H ; A, B ) . Thematrices A ′′ and B ′′ are the same as in (115). Then we can proceed as in [37] and getthe following proposition. he Berry-Keating operator on L ( R > , d x ) and on compact graphs Proposition 13.1 λ = k = 0 is an eigenvalue of ( H ; A, B ) iff for one value k ′ = 0 and then for every k ′ = 0 F ( k ′ ) := det( − S ′′ ( A, B ; k ′ ) C ( a , b ; k ′ )) = 0 (121) is fulfilled with S ′′ ( A, B ; k ′ ) as in (117) and C ( a , b ; k ′ ) := ln (cid:16) b a (cid:17) i k ′ +ln (cid:16) b a (cid:17) . . . ln (cid:16) bEaE (cid:17) i k ′ +ln (cid:16) bEaE (cid:17) i k ′ i k ′ +ln (cid:16) b a (cid:17) . . . i k ′ i k ′ +ln (cid:16) bEaE (cid:17) i k ′ i k ′ +ln (cid:16) b a (cid:17) . . . i k ′ i k ′ +ln (cid:16) bEaE (cid:17) ln (cid:16) b a (cid:17) i k ′ +ln (cid:16) b a (cid:17) . . . ln (cid:16) bEaE (cid:17) i k ′ +ln (cid:16) bEaE (cid:17) . (122) Furthermore, the multiplicity of the eigenvalue λ = 0 coincides with the multiplicity ofthe eigenvalue one of S ′′ ( A, B ; k ′ ) C ( a , b ; k ′ ) for every real k ′ = 0 . Thus, in general there is a difference between the spectral multiplicity of the eigenvalueone of S ′′ ( A, B ; 0) T ( a , b , , which we denote by N , and the eigenvalue one of S ′′ ( A, B ; k ′ ) C ( a , b ; k ′ ) with k ′ = 0 , see [37], [47] and [50].In order to relate the self-adjoint extensions of ( H , D (Γ)) with the self-adjointextensions of the Laplacian − ∆ one has to adjust the lengths as before. However, inoder to attain the same spectrum, except for the case k / ∈ C \ ( ± i σ ( L ′′ ) \ { } ) , one hasto transform the matrices A and B into A ′′ and B ′′ as in (108) and (115). Then thespectrum of the negative Laplacian characterized by A ′′ and B ′′ with the previous choiceof the lengths will coincide with the spectrum of H characterized by A and B in (96).Especially, the functions F ( k ) and F ( k ) in (118) and (121), respectively, will coincidewith the corresponding functions for − ∆ , see e.g. [37] and [43].We remark that the transformation of the matrices A → A ′′ and B → B ′′ andvice versa (actually we consider below the converse direction) cannot be achieved byperturbing the negative Laplacian by a magnetic flux which corresponds to an operatoracting on the edges as (cid:18) dd x j − i A j ( x j ) (cid:19) ψ ( x j ) for ≤ j ≤ E. (123)Kostrykin and Schrader have shown in [51] that this operator is related to the negativeLaplacian − ∆ by a unitary transformation of the corresponding S-matrices. This meansthat the Laplacian perturbed by a magnetic flux can also be characterized by twomatrices A and B obeying (97). But with a local gauge transformation this systemcan be transformed to a quantum graph system with the pure Laplacian which is now he Berry-Keating operator on L ( R > , d x ) and on compact graphs e A and e B . These new matrices are obtained by theold ones by e A = AU and e B = BU (124)where U is a diagonal unitary matrix. If we calculate the S-matrices for these systemswe obtain S ( e A, e B ; k ) = U S ( A, B ; k ) U + . (125)In particular this means that the S-matrix is k -independent iff the original S-matrixis k -independent. By a result of [52] we conclude that in the sense of (109) (see [40])the corresponding matrix P ⊥ ker B LP ⊥ ker B is zero iff P ⊥ ker e B e LP ⊥ ker e B is zero. This feature isobviously not given by the transformation A, B to A ′′ , B ′′ especially in (114) takinginto account that the transformation A, B to A ′ , B ′ in (108) and the correspondingtransformation L to L ′ possess this feature.Kostrykin and Schrader have shown in [44] that the negative Laplacian − ∆ possesses time-reversal symmetry iff S T = S . Obviously, the transformation (125)doesn’t maintain this symmetry in general.In both cases ( H BK and H BK ) we get the same length l i for the edge e i of thequantum graph for the corresponding momentum operator or kinetic energy operator.Thus, we choose l i for the lengths of the graph and endow it with a metric structure asin section 6.
14. Connection between H BK and H BK We also want to reveal the link between H BK and H BK , the last one considered asthe “squared” Berry-Keating operator as in (87). Therefore, we are starting from ( H BK ; A, B ) with corresponding S-matrix S ( A, B ) and then calculate ( H BK ; e A, e B ) withcorresponding S-matrix S ( e A, e B ) . Using the definitions (73) we obtain from (87) the two(necessary and sufficient) equations, setting φ := H BK ψ , A (cid:16) U D ( ab ) Ψ bv (cid:17) + B (cid:16) U D ( ab ) Ψ bv (cid:17) = 0 A (cid:16) U D ( ab ) Φ bv (cid:17) + B (cid:16) U D ( ab ) Φ bv (cid:17) = 0 . (126)Making the definitions ψ ( a ) = ( ψ ( a ) , . . . , ψ E ( a E )) T and ψ ( b ) = ( ψ ( b ) , . . . , ψ E ( b E )) T (127)the equations (126)can be transformed into (see definition (62)) ψ ( a ) = (cid:18) ba (cid:19) S ( A, B ) ψ ( b ) ψ ′ ( a ) = (cid:18) ba (cid:19) S ( A, B ) ψ ′ ( b ) . (128) he Berry-Keating operator on L ( R > , d x ) and on compact graphs − (cid:0) ab (cid:1) S ( A, B )0 0 ! D ( ab ) Ψ bv + B (cid:0) ba (cid:1) S ( A, B ) ! D ( ab ) Ψ ′ bv = 0 . (129)Comparing (129) with (96), we infer for the matrices (one possible choice) e A and e B e A = − (cid:0) ab (cid:1) S ( A, B )0 0 ! , e B = (cid:0) ba (cid:1) S ( A, B ) ! . (130)It is a simple calculation that indeed the matrices e A and e B fulfil (97). In order tocalculate S ( e A, e B ) we make two observations. First, we infer from (106) and (130) that L ′ = 0 in the decomposition (109). Furthermore, we notice that ker e B ′⊥ = span ( S ( A, B ) e i e i ! ; i = 1 , . . . , E ) , ker e B ′ = span ( S ( A, B ) e i − e i ! ; i = 1 , . . . , E ) , (131)where span denotes the linear span of the corresponding vectors, and the vectors e i spanthe space C E . Therefore, by (114) we conclude L ′′ = L ′ = 0 . (132)Thus, by (117) and after an easy (but lengthy) calculation we obtain for the S-matrixof ( H BK ; e A, e B ) S ( e A, e B ) = S ( A, B ) S ( A, B ) + ! . (133)Therefore, we have proved the following proposition. Proposition 14.1 ( H BK ; e A, e B ) is the “squared” operator of some ( H BK ; A, B ) in thesense of (87) iff the corresponding S-matrices fulfil (133). We remark that a similar relation holds for the usual Laplace operator − ∆ and themomentum operator p on compact graphs.
15. Trace formulae and Weyl’s law
We are now in the position to give explicit expressions for the behaviour of the eigenvaluecounting functions for large eigenvalues and give trace formulae for the Berry-Keatingoperator and the “squared” Berry-Keating operator on compact graphs. These resultsare immediate consequences of sections 7, 10 and the results in [37]. The proofs of theclaims for the Berry-Keating operator are quite analogous to [37] and, therefore, we onlygive a short outline of some steps of the proof. Since the trace formulae differ in somedetails we formulate these formulae in one theorem and one corollary. First of all weintroduce an appropriate space of test functions as in [37]. he Berry-Keating operator on L ( R > , d x ) and on compact graphs Definition 15.1
For each r ≥ the space H r consists of all functions h : C → C satisfying the following conditions: • h is even, i.e., h ( k ) = h ( − k ) . • For each h ∈ H r there exists δ > such that h is analytic in the strip M r + δ := { k ∈ C ; | Im k | < r + δ } . • For each h ∈ H r there exists η > such that h ( k ) = O (cid:16) | k | ) η (cid:17) on M r + δ , k → ∞ . We denote by k n the “energies” respectively “wave numbers” of H BK respectively H and by g n the corresponding multiplicities which are identical with the order of thecorresponding zeros k n of F in (83) for n ∈ N respectively zeros k n = 0 of F in (118)for n ∈ N . n = 0 corresponds to the “energy” respectively “wave number” zero, and theenergies respectively the nonnegative wave numbers are ordered with respect to theirabsolute value | k n | in increasing order. However, the (finitely many) imaginary wavenumbers are omitted. Furthermore, we denote the self-adjoint realizations characterizedby (73) respectively (96) by ( H BK ; A, B ) respectively ( H BK ; e A, e B ) . Notice that in thefirst case A, B ∈ Mat( E × E, C ) whereas in the second case e A, e B ∈ Mat(2 E × E, C ) . Inaddition we denote by l min the minimal length of the graph with respect to the definitionof section 6. The minimal positive eigenvalue of L ′′ in (114) is denoted by λ ′′ +min and theunique minimum of the function l ( κ ) := 1 κ ln(2 E ) + 2 κ artanh (cid:18) κλ ′′ +min (cid:19) (134)by σ . For convenience, we denote the total length of the graph by L := E X i =1 l i . (135)Furthermore, by a hat ˆ · we denote the Fourier transform (see (17)) and · ∗ · denotes theconvolution of two functions in the distributional sense, see e.g. [24]. For convenience,we assume that the graph Γ is local with respect to the S-matrix S ′′ ( A, B ; k ) whichmeans that the scattering between two endpoints is only allowed for adjacent edge ends,see [44] for a precise definition. This has the effect that in the trace formula the periodicorbits are with respect to the classical topology as explained in section 6. Otherwisewe must interpret the periodic orbits with respect to the topology induced by the S-matrices which will differ from the one in section 6 and must then be interpreted as aquantum mechanical topology. However, in [44] it was shown that there exists alwaysat least one graph, for which the S-matrix is local. In order to interpret the right sideof (136) for H BK as a sum of periodic orbits, we replace the edges by directed edgesas mentioned in section 6. Furthermore, we assume that the S-matrix S ( A, B ) is localwith respect to the directed edges. This means that S ( A, B ) ij = 0 if e i and e j share novertex v ij for which e j has the direction towards v ij and e i has the direction away from v ij . Again, it is always possible to find such a graph. We get the following theorem for H BK . he Berry-Keating operator on L ( R > , d x ) and on compact graphs Theorem 15.2 (Trace formula for H BK ) Let Γ be a compact metric graph and ( H BK ; A, B ) with the above assumptions be given. Let h ∈ H r with any r ≥ . Then thefollowing trace formula holds (where ˆ h denotes the Fourier transform of h defined as ineq. (17) and g n the multiplicity of the eigenvalue k n ) ∞ X n =0 g n h ( k n ) = L ˆ h (0) + 2 X γ ∈ P Re ( A γ ) ˆ h ( l γ ) . (136)The amplitude functions A γ are constructed from the S-matrix elements with respect tothe periodic orbits γ , see [37, 16] for a precise definition of this construction. The proofof this theorem is quite analogous to [37]. Since we have no k -dependence of S ( A, B ) in (82), we can omit the requirement of the minimal length in contrast to the followingcorollary 15.3. This also leads to the simple product of the real part of the amplitudefunctions A γ and the Fourier transform of h in the identity (136). Furthermore, sincethe secular equation (81) respectively (83) holds also for the eigenvalue zero of H BK , theterm g − N does not appear in (136) in contrast to (137) for H , where N denotesthe multiplicity of the (possible) zero k = 0 of F ( k ) . For H we get the followingtrace formula. Theorem 15.3 (Trace formula for H ) Let Γ be a compact metric graph and ( H BK ; e A, e B ) with the above assumptions be given. Let the condition l min > l ( σ ) befulfilled and let h ∈ H r with r ≥ σ . Then the following trace formula holds ∞ X n =0 g n h ( k n ) = L ˆ h (0) + ( g − N ) h (0) − π Z + ∞−∞ h ( k ) Im tr S ′′ ( A, B ; k ) k d k + X γ ∈ P h (ˆ h ∗ ˆ A γ )( l γ ) + (ˆ h ∗ ˆ A γ )( l γ ) i . (137)Again, the amplitude functions A γ are constructed from the S-matrix elements withrespect to the periodic orbits γ and g denotes the multiplicity of the eigenvalue one of S ′′ ( A, B ; k ′ ) C ( a , b ; k ′ ) for any k ′ ∈ R \ { } (see section 13).Since we have previously seen that the spectrum of ( H BK ; e A, e B ) coincides withsome self-adjoint realization of − ∆ on the graph by adapting the lengths and with theresults in [37], we get Weyl’s law: Theorem 15.4 (Weyl’s law for H ) Given the eigenvalues of some ( H BK ; e A, e B ) in increasing order denoted by λ n = k n . Then for the counting function N ( λ ) := { n ; k n ≤ λ } the following asymptotic law holds N ( λ ) ∼ L π √ λ for λ → ∞ . (138)The same asymptotic law holds for ( H BK ; A, B ) replacing λ by k on the left-hand sideand replacing √ λ by k at the right-hand side on the equation, i.e. we have Theorem 15.5 (Weyl’s law for H BK ) Given the positive eigenvalues of some ( H BK ; A, B ) in increasing order denoted by λ n = k n . Then for the counting function N ( k ) := { n ; k n ≤ k } the following asymptotic law holds N ( k ) ∼ L π k for k → ∞ . (139) he Berry-Keating operator on L ( R > , d x ) and on compact graphs H BK of H BK in the sense of (87) (the spectrum of the eigenvalues of H BK and the wavenumbers of H BK coincides then and therefore the corresponding eigenvalue respectivelywave number counting functions are the same). Comparing the theorems 15.4 and 15.5with the asymptotics of the counting function for the nontrivial Riemann zeros (32), wetherefore can conclude: Theorem 15.6 (No-go theorem)
Neither H BK nor H yields as eigenvalues thenontrivial Riemann zeros if these are self-adjoint realizations on any compact graph.
16. Simple examples
We shall give a simple example for a wave packet and its time-evolution with respectto the Berry-Keating operator in H = L ( R > , d x ) discussed in section 2. Furthermore,we give an example for a realization of H BK and H on the simplest construction of agraph which consists of a single edge. Finally, we present some trace formulae for thepresented examples. Example 16.1
For ψ ( x,
0) = φ ( x ) in (26) we define ( x ∈ R > ) φ ( x ) := α e x + 1 with α = 1 q ln 2 − . (140) (With this choice for α it holds k φ k = 1 .) From (22) we obtain ψ ( x, t ) = ( U ( t ) φ )( x ) = α e − t e x e − t + 1 with t ∈ R . (141) Thus, we get the large- t asymptotics ψ ∼ α − t for t → ∞ . (142) On the other hand, with K BK ( x, x ; t ) = ∞ Z −∞ ψ k ( x ) ψ k ( x )e − i kt d k, (143) (14) and (26), we get ψ ( x, t ) = ∞ Z −∞ A ( k ) ψ k ( x )e − i kt d k. (144) A direct calculation using (19) and the integral representation of ζ ( s ) as a Mellintransform (see [ , p. ) yields A ( k ) = α √ π (cid:16) − √ i k (cid:17) Γ (cid:18) − i k (cid:19) ζ (cid:18) − i k (cid:19) . (145) he Berry-Keating operator on L ( R > , d x ) and on compact graphs With (see [ , p. ) (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:18) − i k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∼ √ π e − π | k | for k ∈ R , | k | → ∞ , (146) we get for the large- k asymptotics of | A ( k ) | | A ( k ) | ∼ α (cid:16) − √ k ln 2) (cid:17) e − π | k | (cid:12)(cid:12)(cid:12)(cid:12) ζ (cid:18) − i k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) for k ∈ R , | k | → ∞ , (147) which gives a sufficient condition for A ∈ L ( R , d k ) . If we consider the continuousrepresentation (144) of ψ ( x, t ) , we see that ψ ( x, t ) gets no contribution from the wavepacket A ( k ) exactly at the wave numbers k corresponding to the conjectured nontrivialRiemann zeros. This is reminiscent to the absorption spectrum interpretation of thenontrivial Riemann zeros by Connes [ , ] , but of course reveals no insight to theposition of the nontrivial Riemann zeros. Example 16.2
For a single edge I = [ a, b ] (one-dimensional quantum billiard) thematrices A and B are arbitrary numbers fulfilling (68). The equations (73) and (82)lead then with S ( A, B ) =: e − π i c (148) to ψ ( a ) = S ( A, B ) r ba ψ ( b )= r ba e − π i c ψ ( b ) with c ∈ [0 , . (149) The eigenvalue spectrum is given by k n = 2 π ln ba ( n + c ) with c ∈ [0 , and n ∈ Z . (150) We now want to calculate H as defined in (87) with (149), in particular the S-matrix and then compare it with the results in section 14. In order to distinguishthe characterizing matrices, we denote these with the subscript · H BK and · H . First,we derive the transformation of A H BK , B H BK into A H , B H for the correspondingoperators related by (87). We get the additional condition ψ ′ ( a ) = (cid:18) ba (cid:19) e − π i c ψ ′ ( b ) with c ∈ [0 , . (151) he Berry-Keating operator on L ( R > , d x ) and on compact graphs The two conditions are equivalent to − (cid:0) ba (cid:1) e − π i c ! Ψ bv + (cid:0) ba (cid:1) e − π i c ! Ψ ′ bv ⇔ − (cid:0) ba (cid:1) e − π i c ! a b ! D ( ab ) Ψ bv + (cid:0) ba (cid:1) e − π i c ! a b ! D ( ab ) Ψ ′ bv ⇔ − (cid:0) ab (cid:1) e − π i c ! D ( ab ) Ψ bv + (cid:0) ba (cid:1) e − π i c ! D ( ab ) Ψ ′ bv . (152) Therefore, we define A H := − (cid:0) ab (cid:1) e − π i c ! and B H := (cid:0) ba (cid:1) e − π i c ! (153) and recognize that indeed A H B + H = B H A + H = 0 and rank (cid:16) A H , B H (cid:17) = 2 (154) is fulfilled. By a comparison of (152) with (96), we infer that A H and B H are twopossible matrices to characterize H in the sense of (96). For S ′′ ( A H , B H ; k ) , weget S ′′ ( A H , B H ; k ) = − π i c e π i c ! = S ( A, B ) S ( A, B ) + ! (155) in complete agreement with (133) and for the secular equation (118) (cid:16) e i ( k ln ( ba ) +2 πc ) − (cid:17) (cid:16) e i ( k ln ( ba ) − πc ) − (cid:17) . (156) This leads to the “wave numbers” k n = 2 π ln ba ( n ± c ) , n ∈ Z (157) with c as in (150). Obviously, with (157) and (150) Weyl’s law is fulfilled even forsmall n . Alternatively, since these are from the classical point of view integrable systemswe can perform an EBK-quantization for H BK and H [ , ] . In this semiclassicalquantization rule the spectrum consists of energies E n (for convenience we use now thesame letter E n for k n respectively λ n as in the sections 2 respectively 4) for which ( ~ = 1 ) I n ( E n ) = (cid:16) n + µ n (cid:17) with n ≥ (158) is fulfilled. Therein µ n denotes the so-called Maslov index and I n ( E n ) = 12 π Z γ n p d x (159) he Berry-Keating operator on L ( R > , d x ) and on compact graphs is the classical action of a periodic orbit γ n which is a subset of the hypersurface H cl = E n respectively e H cl = E n . For H cl in (1) with the ring system structure mentioned in section3, we get from (158) E n = 2 π ln (cid:0) ba (cid:1) (cid:16) n + µ n (cid:17) , (160) and for e H cl in (33) (also with the ring system structure) p E n = k n = 2 π ln (cid:0) ba (cid:1) (cid:16) n + µ n (cid:17) . (161)A comparison of (160) with (150) yields for the Maslov indices µ n = 4 c for H cl . For e H BK we get two Maslov indices, µ n = 4 c for n = 0 , , , . . . and µ n = − c for n = 1 , , , . . . .Since a Maslov index is at most defined modulo and because of c ∈ [0 , , the abovesecond Maslov indices µ n = − c correspond to the Maslov indices ˜ µ n = 4(1 − c ) for n = 1 , , . . . . We stress that the EBK-quantization for e H cl with a classical “hard wall”boundary condition yields p E n = k n = π ln (cid:0) ba (cid:1) (cid:16) n + µ n (cid:17) , (162)which differs from (161) by a factor . We mention that the S-matrix elements of S ′′ ( A H , B H ; k ) for Dirichlet ( D ), Neumann ( N ) or Robin ( R ) boundary conditionsare given by S ′′ (cid:16) A H , B H ; k (cid:17) ij = − δ ij for D , δ ij for N , − δ ij ρ j − i kρ j + i k for R ,with ρ j ∈ R and i, j ∈ { , } , (163)which obviously differ from (155) and, therefore, cannot originate from a “squared”Berry-Keating operator. If we impose Dirichlet boundary conditions at both intervalends, we get for F ( k ) in (118) F ( k ) = 1 − e k ln ( ba ) (164)and thus we obtain for the wave numbers k n k n = π ln (cid:0) ba (cid:1) n, n ∈ Z \ { } , (165)wherein we have taken into account that λ = k = 0 is not an eigenvalue for theDirichlet case. In contrast to the Dirichlet case, λ = k = 0 is an eigenvalue forNeumann boundary conditions at both interval ends, and the nonzero wave numberscoincide with the Dirichlet case (165). For Robin boundary conditions at both intervalends, we get F ( k ) = 1 − ( ρ − i k ) ( ρ − i k )( ρ + i k ) ( ρ + i k ) e k ln ( ba ) . (166) he Berry-Keating operator on L ( R > , d x ) and on compact graphs µ n = 0 for n ∈ N (167)and additionally µ = 0 for the Neumann case. For the Robin boundary conditions onboth interval ends the Maslov indices have to be individually calculated for each n ∈ N by (166).The above considerations underline the fact that the form of the S-matrix (155)corresponds to a pure ring system as in the case of the negative Laplacian − ∆ , see [ ] . The occurrence of possible noninteger “Maslov indices” originates simply from thefact that we have a discontinuous crossover by turning once around in the ring system(one-dimensional torus) in contrast to the “usual” continuity requirement of the wavefunction, see e.g. [ ] . Example 16.3
We shall present a trace formula for the time-evolution operator U ( t ) in (21) and (22) for H BK acting on a single edge with the assigned interval I = [1 , b ] : ( U ( t ) φ ) ( x ) := ∞ X n = −∞ ψ n ( x ) h ψ n , φ i e − i k n t , φ ∈ L ( I, d x ) , (168) with the eigenvalues k n = π ln b ( n + c ) , n ∈ Z , c ∈ [0 , , and the normalized eigenfunctions ψ n ( x ) = 1 √ x ln b e i k n ln x with n ∈ Z . (169) For the corresponding (not retarded) integral kernel of U ( t ) we get by [ , p. (in a distributional sense acting on D ( I ) ⊂ L ( I, d x ) identified by the continuousrepresentatives; g ( x, x ; t ) := π ln b [ln x − ln x − t ] ) K ( x, x ; t ) := ∞ X n = −∞ ψ n ( x ) ψ n ( x )e − i k n t = e i cg ( x,x ; t ) √ xx ln b ∞ X n = −∞ e i g ( x,x ; t ) n = 2 π ln b e i cg ( x,x ; t ) √ xx ∞ X n = −∞ δ ( g ( x, x ; t ) + 2 πn )= e i cg ( x,x ; t ) ∞ X n = −∞ b n e t δ (cid:0) xb n − x e t (cid:1) . (170) If we take the trace of U ( t ) , we obtain with (170) (by defining the “period” T = ln b [see(3)] and the Maslov index µ := 4 c ) Tr U ( t ) := b Z K ( x, x ; t )d x = ∞ X n = −∞ e − i k n t = T ∞ X n = −∞ e − i π µn δ ( t − nT ) . (171) he Berry-Keating operator on L ( R > , d x ) and on compact graphs If we now choose a test function h of H r (definition 15.1) with an arbitrary r > , weget by the identity (171) and the symmetry of h the trace formula ∞ Z −∞ ˆ h ( t ) Tr U ( t )d t = ∞ X n = −∞ h ( k n )= T ˆ h (0) + T ∞ X n =1 (cid:16) e − i π µn ˆ h ( nT ) + e i π µn ˆ h ( − nT ) (cid:17) = T ˆ h (0) + 2 T ∞ X n =1 cos (cid:16) π µn (cid:17) ˆ h ( nT ) . (172) We recall that the S-matrix for this quantum graph is S ( A, B ) = e − π i c = e − i π µ (see(148)), and the length of the (single) edge is l = L = ln b = ln b . Since we havea directed edge, there is only one possibility for the orientation of the periodic orbitsand, therefore, the periodic orbits can be labelled by the natural numbers, and thecorresponding lengths of the periodic orbits are l n = n ln b and all are multiples of oneprimitive periodic orbit with length l = ln b = T . For the amplitude functions we get(see [ ] ) A n = l e − π i cn = T e − i π µn . Applying (136) we get (172), which confirms thetrace formula in theorem 15.2. Example 16.4
We shall present a trace formula for the kernel e K ( x, x ; t ) of the unitaryevolution operator e − i tH of H with Dirichlet boundary conditions ( D ) on a singleedge e with assigned interval I = [1 , b ] . The eigenvalues are given by (165), thus the(Feynman-)kernel reads e K ( x, x ; t ) := ∞ X n =1 ψ n ( x ) ψ n ( x )e − i k n t (173) with the normalized eigenfunctions ψ n ( x ) := r l sin (cid:0) nπ ln x l (cid:1) √ x , n ∈ N , (174) where l := ln b denotes the length of the edge e . Using a suitable addition theoremfor trigonometric functions, we get two alternative expressions for e K ( x, x ; t ) (seehe Berry-Keating operator on L ( R > , d x ) and on compact graphs [ , p. e K ( x, x ; t ) = 1 √ xx l γ p " Θ l γ p ln (cid:18) xx (cid:19) , − π l γ p t ! − Θ l γ p ln ( xx ) , − π l γ p t ! = 12 √ xx √ i πt ∞ X n =0 ǫ n " e i2 nπ exp i (cid:0) ln x − ln x + n l γ p (cid:1) t ! +e i(2 n +1) π exp i (cid:0) ln x + ln x + n l γ p (cid:1) t ! + e i2 nπ exp i (cid:0) ln x − ln x − n l γ p (cid:1) t ! +e i(2 n +1) π exp i (cid:0) ln x + ln x − n l γ p (cid:1) t ! , (175) where Θ ( z, τ ) denotes the Jacobi theta function and we have defined ǫ n := for n = 01 for n > . (176) l γ p := 2 l = 2 ln b is the length of the primitive periodic orbit γ p of the correspondingclassical system. Notice that the summands in the second identity in (175) can beinterpreted as contributions of free particle kernels at a fixed time t corresponding tothe four types of paths p ( x , x ) (see section 6) joining x and x (see e.g. [ , ] ). Forthis reason, we define (cf. (42) and [ , p. e K p ( x, x ; t ) := 12 √ i πt exp (cid:18) i l p ( x , x ) t (cid:19) (177) where l p ( x , x ) is the length of the path p ( x , x ) (see (49)), and we then get by (175) e K ( x, x ; t ) = 1 √ xx X p ( x ,x ) exp (cid:0) i πn p ( x ,x ) (cid:1) e K p ( x, x ; t )= 1 √ xx X p ( x ,x ) exp (cid:16) − i πµ p ( x ,x ) (cid:17) e K p ( x, x ; t ) , (178) where the sum comprises all possible paths p ( x , x ) joining x and x , and n p ( x ,x ) isdefined as the number of reflections of the path p ( x , x ) at the “hard wall” intervalendpoints and ln b . µ p ( x ,x ) denotes the Maslov index of the path p ( x , x ) which isgiven by µ p ( x ,x ) = 2 n p ( x ,x ) mod 4 in agreement with the “usual” Maslov index for theone-dimensional billiard system corresponding to the negative Laplacian (see [ ] ), andwith (167) (in (167) the Maslov index corresponds to periodic orbits). Example 16.5
Finally, we shall present an explicit trace formula (heat kernel) for asingle edge with assigned interval I = [1 , b ] for H with Dirichlet boundary conditions ( D ) . We calculate the trace of the heat kernel of e − tH (replacing t by − i t in (173)) e K h ( x, x ; t ) := e K ( x, x ; − i t ) = X k n ψ ( x ) ψ n ( x )e − k n t (179) he Berry-Keating operator on L ( R > , d x ) and on compact graphs directly and then compare the result with the trace formula (137). Therefore, we recallthat the wave numbers of H with ( D ) are explicitly given by (165). Thus, we obtainfor the trace of the heat kernel (setting L := l γ p := ln b and [ , p. ) Tr e − tH = b Z X k n ψ ( x ) ψ n ( x )e − k n t d x = X k n e − k n t = 12 Θ (cid:16) , i 4 π l γ p t (cid:17) − ! = l γ p √ tπ ∞ X n = −∞ exp (cid:16) − n l γ p t (cid:17)! − L √ πt −
12 + ∞ X n =1 l γ p √ πt e − ( n l γp ) t . (180) Notice that the sums in (180) are absolutely convergent whereas in (171) the sums areconvergent in the topology of D ′ ( R ) (in a distributional sense). In order to compare thisresult with (137), we recall that C (1 , ln b ; k ′ ) and the S-matrix S ( D ) for the Dirichlet caseis given by (see (121) and (163)) S ′′ ( D ) = − × and C (1 , ln b ; k ′ ) = ln b i k ′ +ln b i k ′ i k ′ +ln b i k ′ i k ′ +ln b ln b i k ′ +ln b . (181) It is a simple calculation that the multiplicity g of the eigenvalue one of S ′′ ( D ) C (1 , ln b ; k ′ ) is g = 0 for any k ′ ∈ R \ { } . Furthermore, it is obvious that the order N of the zerowith wave number k = 0 of F ( k ) in (118) is N = 1 , thus g − N = − . Themultiplicities of the wave numbers k n are g n = 1 for n ∈ N . Since a Dirichlet boundarycondition corresponds to the classical “hard wall” boundary condition, we conclude thatthe periodic orbits γ are given by all multiples of one primitive periodic orbit γ p withprimitive periodic orbit length l γ p = 2 ln b . For the amplitude functions A γ in (137)we obtain A γ = l γ p (see [ ] ). Furthermore, it is obvious that Im S D = 0 . Using thetest function h ( k ) := e − k t we obtain the Fourier transform ˆ h ( x ) = √ πt e − x t . Insertingthese quantities in the trace formula (137) we get the trace formula (180), which againconfirms the trace formula (137). We remark that from the small- t asymptotics (180)one obtains directly the Weyl asymptotics (138) using a proper Karamata-Tauberiantheorem.
17. Summary and conclusions
We have studied the quantization of the extraordinarily simple classical Hamiltonian H cl ( x, p ) = xp about which Berry and Keating [13, 14] speculated that some quantizationof it might yield the hypothetical Hilbert-Polya operator [1, 2, 3, 4, 10, 11, 12] possessingas eigenvalues the nontrivial Riemann zeros. Two quantum Hamiltonians respectively he Berry-Keating operator on L ( R > , d x ) and on compact graphs H BK := − i ~ (cid:0) x dd x + (cid:1) obtained from H cl by Weyl ordering, and the so-called “squared”Berry-Keating operator H := − x d x − x dd x − which is a special case of the famousBlack-Scholes operator [33, 34] used in the financial theory of option pricing.In section 2, we have given a complete description of the quantum dynamicsgenerated by H BK acting in the Hilbert space L ( R > , d x ) . While the one-dimensionalquantum system governed by H BK possesses many interesting properties, one of our mainresults of section 2 is that the spectrum of H BK is purely continuous corresponding toscattering states. Since there are no bound states corresponding to a discrete spectrum,it is obvious that this specific quantization of the Berry-Keating operator cannot possessthe Riemann zeros as part of its spectrum. Let us point out, however, that in the simpleexample 16.1 we have studied the quantum dynamics of H BK for the particular square-integrable wave function (141) for which it turns out that the spectral decompositionconsists of a continuous wave number spectrum, k ∈ R , into which there are embeddedinfinitely many absorption lines located exactly at the wave numbers k n = τ n ∈ R corresponding to the nontrivial Riemann zeros satisfying the Riemann hypothesis. Toour knowledge, there is, however, no relation of this occurrence of the Riemann zeros tothe absorption spectrum interpretation of Connes [11, 12].Analogous results have been obtained in section 4 for the “squared” Berry-Keatingoperator H acting in the Hilbert space L ( R > , d x ) . We have proved that in this casethe spectrum is purely continuous too and thus there holds again a “no-go theorem”with respect to the identification of H with the hypothetical Hilbert-Polya operator.In the main part of our paper, we have dealt with the quantum dynamics of H BK respectively H on compact quantum graphs introduced in section 6. After havingdefined the Berry-Keating operator H BK on compact graphs in section 7, we havegiven in proposition 8.1 a complete classification of all self-adjoint extensions of H BK oncompact quantum graphs in terms of two matrices A and B satisfying the conditions(68). In proposition 9.1 we have established a secular equation valid for any self-adjointrealization in form of a determinant whose zeros determine the discrete spectrum of H BK .In the sections 10 -13, we have studied the quantization of the “squared” Berry-Keating operator H on compact quantum graphs. Proposition 11.1 provides thecomplete classification of all self-adjoint realizations of H again in terms of matrices A and B satisfying in this case the conditions (97). For the discrete spectrum of H ,we have given in proposition 12.1 the corresponding secular equation for λ = k = 0 .The zero eigenvalue λ = k = 0 of H plays a special rôle which we have discussed insection 13 leading to the additional secular equation (121). Furthermore, in section 14we have discussed the conditions under which H is the square of H BK in the sense of(87), see proposition 14.1.Based on the results derived in the previous sections, we have been able in section15 to state several theorems. In the theorems 15.2 and 15.3 we have given an exact traceformula for H BK respectively H for a large class of test functions h ( k ) belonging to he Berry-Keating operator on L ( R > , d x ) and on compact graphs H r defined in definition 15.1. The trace formulae establish a deep connectionbetween the eigenvalue spectra of H BK respectively H and the length spectra of theperiodic orbits of the corresponding classical dynamics.As an important consequence of the trace formulae, we have derived the Weylasymptotics for H BK (for H we have used results in [37]). The Weyl asymptoticsof these operators have been given in the theorems 15.5 and 15.4, respectively. Acomparison with the expected Weyl asymptotics (48) respectively (32) for the nontrivialRiemann zeros demonstrates clearly that neither H BK nor H can yield as eigenvaluesthe nontrivial zeros of the Riemann zeta function if these operators are self-adjointrealizations on any compact quantum graph, see theorem 15.6.Finally, we have presented in section 16 four simple examples illustrating someaspects of the quantum dynamics of H BK respectively H . Acknowledgements
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