The properties of the distribution of Gaussian packets on a spatial network
aa r X i v : . [ m a t h - ph ] N ov The properties of the distribution of Gaussian packets on a spatialnetwork.
V. L. Chernyshev, A. A. TolchennikovNovember 6, 2018
Abstract
The article deals with the description of the statistical behavior of Gaussian packets on a metric graph.Semiclassical asymptotics of solutions of the Cauchy problem for the Schr¨odinger equation with initial dataconcentrated in the neighborhood of one point on the edge, generates a classical dynamical system on a graph.In a situation where all times for packets to pass over edges (”edge travel times”) are linearly independentover the rational numbers, a description of the behavior of such systems is related to the number-theoreticproblem of counting the number of lattice points in an expanding polyhedron. In this paper we show thatfor a finite compact graph packets almost always are distributed evenly. A formula for the leading coefficientof the asymptotic behavior of the number of packets with an increasing time is obtained. The article alsodiscusses a situation where the times of passage over the edges are not linearly independent over the rationals.
In this paper we study semiclassical solutions of the Cauchy problem for a time-dependent Schr¨odinger equationon a metric graph. This article is, in a s certain sense, a continuation of [2], [3] where you can find necessarydefinitions and formulations (nevertheless, we quote all necessary terms here, in section 1.2). There (and in [4])are also references to some articles and reviews related to the study of differential equations on metric graphs.In [3] a formula for the asymptotics of the number of Gaussian packets was written for a star graph, and aquestion of finding the leading coefficient for the case of an arbitrary finite compact graph was raised. In thispaper that problem is solved and a general formula for that coefficient is obtained (see Theorem 2.2). Moreover,in [3] the uniformity of distribution of packets for the special case of two vertices connected by three edgeswas proven. In this paper we present the proof of uniformity of the asymptotic distribution of packets over anarbitrary finite compact graph for almost all edge travel times (see Theorem 2.1).In the second part (section 3) we discuss a description of the asymptotic behavior of packets in the caseof linearly dependent over Q edge travel times. In this situation there is no correspondence (see [2], [3] fordetails) between the number of packets and the number of nodes of an integer lattice that lie on some faces ofan expanding simplex, but analysis of their number is still possible. If rank of the system of edge travel times(lengthes) is equal to one then the number of packets grows only in a finite interval of time and stabilizes at acertain value, which depends on the lengths of cycles. Also we consider an example of a graph, whose rank ofthe system of lengthes equals two. V. L. Chernyshev thanks M. M. Skriganov, N. G. Moschevitin, P. B. Kurasov and O. V. Sobolev for useful discus-sions and attention to his work. Authors are grateful to A. I. Shafarevich for constant attention to their work.The work is done with partial finance support of grants MK-943.2010.1, RFFI 10-07-00617-a and 09-07-00327-a,RNP 2.1.1/11818 and state contract 14.740.11.0794.
Recall (see, e.g., [8], [9] and references therein), that a metric graph is a one-dimensional cell complex whoseedges are parameterized curves. We denote a geometric graph by Γ, its edges by γ j , its vertices by a j . A setof all edges adjacent to the vertex a we denote by Γ( a ). We consider only finite metric graphs. Let E and V stand for the number of edges and the number of vertices respectively.1et Q be an arbitrary real valued continuous function on Γ, smooth on the edges. Schr¨odinger operator b Hψ = − h d ψ ( x ) dx + Q ( x ) ψ ( x ) (1)is defined on the set of functions from Sobolev spaces ψ ∈ ⊕ P j H ( γ j ), satisfying the following boundaryconditions at the vertices:1. function ψ is continuous on Γ;2. X γ j ∈ Γ( a m ) α j dψ j dx ( a m ) = 0 , (2)in all internal vertices (i.e., the vertices of degree greater than one);3. ψ ( a m ) = 0 in all the external vertices, i.e. vertices of degree one.Here α j = 1 for each edge emerging from the vertex, and α j = − natural (see [10]). They, in particular, ensure self-adjointness of the operator b H . A time-dependent Schr¨odinger equation on the graph
Γ is an equation of the form ih ∂ψ∂t = b Hψ, (3)where a semiclassical parameter h →
0. We choose initial conditions that have the form of a narrow packetlocalized near the point x , which lies on the edge of the graph: ψ ( x,
0) = h − / K exp (cid:18) iS ( x ) h (cid:19) . (4) S ( x ) = a ( x − x ) + b ( x − x ) + c, where b and c are real constants, and a and K are complex. The imaginary part of a is positive. Normalizationfactor h − / is introduced to ensure that the initial function ψ ( x,
0) is of order one in the norm of L (Γ). Dueto the positivity of the imaginary part of a the initial function is localized in a small neighborhood of x : ψ ( x,
0) = O ( h ∞ ) with | x − x | ≥ δ > δ is independent of h ). For simplicity we assume that there are noturning points (see, e.g., [7], [6]) on the Γ ; their presence can be accounted for in the standard way (see [7]).Asymptotic solution of the Cauchy problem (3)–(4) is described in [3], the explicit formulas are given therein. Theorem 1.1 (See [3]) Solution of the Cauchy problem 3 – 4 for t ∈ [0 , T ] ( T does not depend on h ), is givenby the following formula ψ ( x, t, h ) = N ( t ) X j =1 h − / ϕ j ( t ) e iS j ( x,t ) /h + O ( √ h ) , (5) where the functions ϕ j ( t ) , S j ( t ) are explicitly expressed in terms of the solutions of two hamiltonian systems. Each term in the sum (5) is localized in a small neighborhood of X j ( t ). Here we assume that all the termsthat are localized in the same point X j ( t ) form one Gaussian packet . Later, under the N ( t ) we would mean thenumber of such packets.In [3] it is shown that in moments of penetration of the vertices of the graph a quantum packet is dividedinto m packets ( m is a degree of the vertex): one is reflected and m − m −
2) : 2 (2 for the scattered and ( m −
2) for the reflected packet).We consider the asymptotical behavior of function ψ ( x, t h ) as t →∞ . Namely, we will see how the numberof Gaussian packets N ( t ) changes in time. Note that this problem differs from the task of describing theasymptotic solution of the Schr¨odinger equation at t →∞ , as the error estimation is valid only for finite times.From a physical point view, this means that we are considering big t , but much smaller than 1 /h .Let t j stand for j -th edge travel time (this is an analog of length).2 emma 1.1 Travel time of any edge of the graph depends only on the initial data and is the same for anyGaussian packet on each fixed edge.
Proof.
Let us take an edge with index j . By construction a solution on each edge (using the method ofMaslov complex germ), we find that P + Q ( x ) = E j holds (by the energy conservation law for Hamiltoniansystems), where the value of E j is determined by the initial conditions. Writing P as dXdt , we obtain for edgetravel time on the edge an explicit expression t j = b R a dx √ E j − Q ( x ) , where the integral is taken over the edge. Thevalue of E j is the same for all edges, as the potential Q ( x ) is assumed to be continuous function in the vertices,and P can only change sign (as demonstrated by a construction of the solution). So the formula E j = P + Q ( x )determines the same value E j = E for all packets. At the initial moment of time E = (cid:0) ∂S ∂x (cid:1) + Q ( x ). Valuesof t j may be different, since the restriction of the potential Q ( x ) on the edges may be different. Definition.
Let us consider the number of packets coming out of a fixed vertex to a fixed edge. The leadingcoefficient of the asymptotics of this number is called a radiation coefficient.Correctness of this definition will be shown in the proof of the theorem 2.1. Q edge travel times. Theorem 2.1 (About uniformity of distribution)
Consider a finite connected graph Γ . Suppose that forany vertex its degree is not equal to two. Suppose that there are no turning points on edges. Consider an edge e . Let F e ( t ) be a ratio of N e ( t ) (number of packets on the edge e ) to N ( t ) (total number of packets). Then foralmost all incommensurable (i.e. linearly independent over Q ) numbers t , . . . t E , a ratio of F e ( t ) to the lengthof e tends to a constant E P j =1 t j ! − as time increases. Remark 2.1
It means that the distribution of number of packets tends to a uniform distribution as time in-creases.
Proof.
Let us choose on any edge with travel time t j a segment dg with travel time τ . Let us find N τ ( t ) /N ( t ).We know (see [3]) that N ( t ) = Ct E − + o ( t E − ). Let us find N τ ( t ). Since the number of packets changes onlyin vertices and there are no turning points, then: N τ ( t ) = N → d ( t ) − N → d ( t − τ ) + N → g ( t ) − N → g ( t − τ ) . (6)Here N → d ( t ) stands for the number of packets which arrived at the edge from point d .It is clear that the number of packets arrived at point d at time t equals the number of packets, which cameout from the nearest vertex a at time t − T . Here T is a travel time from a to d . By N a → d ( t ) we denote thenumber of packets, which came from a to d .We have to know asymptotics of the number of packets that come out of a vertex a . Packets can come outof the vertex only at times that are linear combinations (with nonnegative integer coefficients) of edge traveltimes.The number of release moments (when at least one packet comes out of the vertex a ) is described by thenumber of set { n j } satisfying inequations of a kind: n t l + . . . + n m t l m ≤ t, (7)where t j is a travel time of the j -th edge.Since the leading part of asymptotics of the number of packets is defined by the volume of a simplex definedby (7), events with maximal numbers of summands happen more often. In other words, packets arrived at ourvertex should visit all edges. I.e. N a → d ( t ) = R a t E + o ( t E ) . (8)For almost all t , . . . t E the estimation can be improved (see [1]). There exists K a such that N a → d ( t ) = R a t E + K a t E − + o ( t E − ). Let us show that R a , which is called a radiation coefficient , does not depend on the3hoice of a vertex. Consider vertices a and b . There exists a path connecting a and b . Let δ be its travel time.Any packet coming out from a to d over time that does not exceed 2 δ generates at least one packet that comeout from b to d ′ . This is correct for packets coming out from b . We obtain inequations: N a → d ( t + 2 δ ) ≥ N b → d ′ ( t )and N b → d ′ ( t + 2 δ ) ≥ N a → d ( t ). We know that N a → d ( t ) = R a t E + o ( T E ), N b → d ′ ( t ) = R b t E + o ( T E ). Thus R a t E + o ( t E ) = R b t E . Hence R a = R b .Let us modify the expression for N τ ( t ) N τ ( t ) = R ( t − T ) E + K a ( t − T ) E − − R ( t − T − τ ) E − K a ( t − T − τ ) E − + R ( t − T ) E + K b ( t − T ) E − − R ( t − T − τ ) E − K b ( t − T − τ ) E − + o ( t E − ) = 2 ERτ t E − + o ( t E − ).Thus we obtain N τ ( t ) N ( t ) → ERC τ. (9)It remains to show that a coefficient near τ has the required form.We consequentially take edges as dg and then summarize obtained expressions:1 = E X j =1 N t j ( t ) N ( t ) → ERC E X j =1 t j . (10)Hence, C = 2 ER E X j =1 t j . (11)The proof is completed. Remark 2.2
For any two vertices numbers K a and K b are related with inequation | K b − K a |≤ EδR . Here δ is a travel time for any path from a to b . In the proof we obtain the following statement:
Consequense 2.1 (Relation between coefficients C and R ) The leading coefficient for the number of pack-ets C and the radiation coefficient R , for almost all edge travel times, are related in the following manner: C = 2 ER E X j =1 t j . (12) Theorem 2.2 (About the leading coefficient of the number of packets)
Consider a finite connected com-pact graph Γ . Suppose that there are no vertices of degree two. Suppose that there are no turning points onedges. Then for almost all incommensurable numbers t , . . . t E the leading coefficient has the following form: C = 12 V − ( E − E P j =1 t jE Q j =1 t j . (13) Proof is based on (13) and the following lemma.
Lemma 2.1
Let us consider a finite connected graph with incommensurable edge travel times t i ( i = 1 . . . E ) and a number of independent cycles β . Let B be an arbitrary vertex. Then for almost all edge travel times thenumber of packets arriving at B at time T asymptotically equals R ( T ) ∼ β E E ! E Q j =1 t j T E . emark 2.3 This is equivalent to R = 12 V − E ! 1 E Q j =1 t j . Proof.
Is is sufficient to consider paths of packets that traveled upon all edges. Only those paths give usthe leading coefficient. Let A be an initial vertex. For each such path we can construct a “code” i.e. a sequenceof coefficients of a corresponding chain with coefficients in Z . It is clear that the code does not change undera path homotopy. Let us find the number of all possible codes. Consider cross connections, i.e. edges that arenot in the spanning tree. Parity of passages on cross connections defines a path’s homotopy class. All chaincoefficients are defined by coefficients on cross connections. Thus the number of possible codes equals 2 β . Nowfor every code ( c , . . . , c E ) , c i ∈ { , } we associate times ( E X i =1 t i ( c i + 2 n i ) | n i ∈ N ∪ { } ) . At every such time (they are different) at least one packet arrives at the vertex B . The number of such timesthat are less than T asymptotically equals to T E E E ! t · · · t E . Finally we summarize this over all possible codes. The proof of the lemma is finished. At the end we applyEuler’s relation β = E − V + 1 (see, for example, [5]). In this section we assume that travel times are linearly dependent over Q . It means that there is no one-to-onecorrespondence (described in [2], [3]) between the number of packets and the number of integer lattice pointsin an expanding simplex.For the simplest example consider a star graph with three edges of the same length. The number of packetsreaches three and does not increase. While the number of integer lattice points grows with time. Statement 3.1
Consider a finite graph with edge travel times t , = n t . . . , t E = n E t , where n i ∈ N andGCD ( n , . . . , n E ) = 1 .Then starting from a certain time, the number of packets becomes constant.1) If there exists a cycle with travel time kt , k ∈ N then N ( T ) = 2 E X i =1 n i .
2) Otherwise N ( T ) = E X i =1 n i . We omit the proof.
Statement 3.2
Consider a star graph with three edges e , e , e with travel times t = nt , t = mt , t , where n ∈ N , m ∈ N (GCD(n,m) = 1), and t is such that rank { t , t , t } over Q equals 2. Then the number ofpackets asymptotically equals N ( T ) = T (cid:18) m + nt + 1 t (cid:19) + o ( T ) . roof. The number of packets changes only at times of a kind 2(( nα + mβ ) t + t γ ), where α, β, γ . Notethat this representation is not unique. If we can present a time moment with α ≥ , β ≥ , γ ≥
1, then at thatmoment N ( T ) does not change (since packets arrive at a vertex from all three edges) Remark 3.1
Number nα can be represented as nα ′ + mβ ′ ( β ′ = 0 ) if and only if α ≥ m . Hence to define asymptotics of N ( T ) it is sufficient to consider the following times:1) γ = 0 , β = 0 , ≤ α < m . At those times N ( T ) increases by 1. The number of such times equals the numberof pairs ( α, γ ) such that 2( αt + γt ) < T . The number asymptotically equals: ( m − T t .2) γ = 0 , α = 0 , ≤ β < n . Similarly the number of pairs asymptotically equals ( n − T t .3) α = 0 , β = 0. The number of such times that are less than T increases as T t . Here the number of packetsincreases by 2.4) γ = 0. Let us show that linear combinations αn + βm contain all natural numbers that are greater than acertain fixed number. Lemma 3.1 If ( m, n ) = 1 then there exists M such that any N > M can be presented as N = αn + βm , where β ≥ , α ≥ . Proof.
Numbers m, m, . . . , nm give all residues modulo n : { βm } nβ =1 = { β i m = i + nk i | i ∈ [0 , n − , k i ∈ Z + } . Let k = max k i . Choose any N > nk . Then N = i + nk ( k > k , ≤ i < n ) and N − β i m = n ( k − k i ). Thusany number that is greater than nk can be represented as αn + βm , where 1 ≤ β ≤ n, α ≥ kt ( k ∈ N ) that are lessthan T . Since at such times 2 packets arrive from e and e , then N ( T ) increases by 1. References [1] Skriganov M. M. Ergodic theory on SL( nn