# Effective Schrödinger dynamics on -thin Dirichlet waveguides via Quantum Graphs I: star-shaped graphs

###### Abstract

We describe the boundary conditions at the vertex that one must choose to obtain a dynamical system that best describes the low-energy part of the evolution of a quantum system confined to a very small neighbourhood of a star-shaped metric graph.

###### ams:

81Q35, 81Q37^{†}

^{†}: J. Phys. A: Math. Gen.

,

in memory of Pierre Duclos

## 1 Introduction

Often in Physics at the nanoscale one deals with systems which can be regarded as having support in a very small neighborhood of a one dimensional graph . The parameter is a measure of the distance of from as compared to the length of a typical edge of the graph.

The dynamics of such systems is described by a propagation equation, which can be the wave equation in the case of optical wires or the Schrödinger equation in the case of conduction electrons in a macromolecule or in the case of conducting nano-devices. In the case of optical wires it is reasonable to choose Neumann (reflecting) boundary conditions at the boundary . In the case of macromolecules or nano-devices on the other hand one can consider that the system is confined in the region by very strong forces at the atomic scale, which provide a potential having the form of very deep and narrow valleys. In this case the use of Dirichlet boundary conditions can be regarded as a reasonable approximation.

The inital value problem is well posed for both boundary conditions, but in general one is not able to give the solution in an explicit form. It is then useful to search for a dynamical system which is explicitly solvable and at the same time provides an approximation to the physical one when is very small. In this case it is important to give an estimate of the error one makes in this approximation for the observables of interest.

One can expect that the limit dynamical system be described by a wave equation or respectively a Schrödinger equation on the graph. But if this is the case, in order to have a well posed problem one must specify boundary conditions at the vertices. So the problem becomes: which are the boundary conditions (if any) that one must choose at the vertices of the (metric) graph, in order to have an evolution equation on the graph with solutions which give an approximation (to order , for some ) for the expectation values of the relevant physical observables? One may reasonably expect that the answer depends on the shape of in the vicinity of the vertices.

In the case of Neumann boundary conditions the answer is known [exner05] for the case of the Schrödinger equation, at least for initial data with not too large energy. Indeed in this case the initial wavefunction can be chosen smooth (e. g. in ) uniformly in and the solution can be restricted to the graph uniformly when . As a consequence, one can define a sequence of maps from to , and prove that the trace on of the resolvent of the Schrödinger operator on converges to the resolvent of a Schrödinger operator on with Kirchhoff (coupling) vertex conditions (continuity of the solution, zero sum of the outward derivatives). The situation is entirely different in the case of Dirichlet boundary conditions on : the -norm of the initial datum increases without bound as , and there is no limit trace on . For this reason, the question of the existence of a limit flow had not been answered so far [kuchment08], in spite of the obvious physical and mathematical interest.

In concrete cases of physical devices, models of limit dynamics have been constructed to fit the experimental data. For instance, in the case of a sharply bent conducting device experimental evidence shows that in order to have a good approximation one must use Dirichlet (decoupling) boundary conditions at the bend. On the other hand, the standard treatment of conducting electrons in aromatic molecules (such as graphene or benzene), in which the molecule is represented by a graph, shows that results in fair accordance with experiments are obtained if the limit model is constructed with conditions at the vertices that are of weighted Kirchhoff type. The phenomenological values of the weights are different for different molecules. One can consider this as an evidence that the “right” boundary conditions depend on the shape of .

Indeed the images seen at the electronic microscope are different for different molecules (one takes to be the region in which the density of conducting electrons is appreciably different from zero) but in all cases they have the shape of an annulus around the nucleus, with tunnels in correspondence of the valence bonds. One interprets this structure as due to the combined action of the attraction by the nucleus and the exclusion principle that forbids the conduction electrons to occupy the region of the core electrons.

Here we give an answer to the mathematical question described above. We show that the generator of the limit dynamics on the graph corresponds to a suitable boundary condition at the -th vertex , which depends on the shape of near through the spectral properties of a sequence of auxiliary Schrödinger operators defined on a suitable neighbourhood of the vertex .

In this paper we analyze the problem of approximating the free Schrödinger evolution on an -thin Dirichlet star-shaped waveguide, with semi-infinite cylindrical ends of thickness ; in a future publication we shall generalize our results to cover generic waveguides and smooth potentials. In this paper we study convergence of solutions and (weak) convergence of resolvents, and briefly suggest how to add an external potential and study convergence of scattering matrices.

## 2 Description of the problem

In this section, we describe the structure of the waveguide in detail, and we introduce some notation. The waveguide is obtained by gluing smoothly to the vertex region (a compact set in ) a number of semi-infinite cylindrical ends, the branches of the waveguide; the boundary is smooth^{1}^{1}1For illustrative purposes, in the Figures we depict two-dimensional waveguides with edges at some points of the boundary. Note that the hypothesis of smooth boundary could be weakened, and the dimension of the waveguide could be any .. The cylindrical branches have two-dimensional section , a compact domain in with smooth boundary, of linear dimension proportional to . Also, the linear dimension of the vertex region is proportional to so that the waveguides associated to different values of are connected by a similarity transformation (a change in the length scale). With this assumption, the geometry of is essentially fixed, and the problem is equivalent to the computation of the low-energy effective dynamics in a fixed domain. Note that this is not true anymore when we have many vertices at a fixed distance, and we shrink the cylinders connecting them.

### Structure of the waveguide

The branches of the waveguide are labeled by the index , which runs from to . Each branch is isometric to the cylinder ; then

The Dirichlet Laplacian on has discrete spectrum

and the eigenvalues satisfy the scaling relation

The Dirichlet Laplacian on the entire waveguide has absolutely continuous spectrum which coincides with the semi-infinite interval , and possibly some discrete eigenvalues , not larger than , corresponding to bound states^{2}^{2}2bound states are known to form e.g. near the bent regions of the waveguides; see for example [exner89] with eigenfunctions . They also satisfy the scaling rule

It is natural to work with the renormalized Hamiltonian

acting on . In this way, the bound states have negative energy and the absolutely continuous spectrum coincides with the positive half-line.

Let us introduce the crucial concept of mesoscopic region , or sometimes simply . This is obtained by cutting at distance from the vertex region :

The complement of is referred to as :

It is very convenient to work with the parameter defined as

instead that with . We refer to as to the rescaled mesoscopic lenght. With this notation, the mesoscopic region has tickness and size .

Remark To identify the points in the -th branch of the waveguide, we normally use Cartesian coordinates , with values in . We can say that these coordinates are adapted to the macroscopic scale. Later it will be convenient to make use also of rescaled coordinates , adapted to the microscopic scale, ranging in , such that

Notice that in these coordinates, the mesoscopic region is characterized by the inequalities

##### The general strategy

The waveguide that we consider is characterized by two parameters: the microscopic lenght and the mesoscopic lenght . The parameter is chosen so that the inequality

is satisfied: this motivates our notation, too. In terms of the rescaled mesoscopic lenght , these conditions read

Our strategy is the following: given a wavefunction on , solution of the Schrödinger equation, we consider its restriction to and . We prove that essentially factors into plus negligible corrections: is the effective wavefunction on the limit graph . It is clear that this approach makes sense if , so that is defined everywhere except for a small neighbourhood of the vertex.

Our second step is to understand the behaviuor of near the vertex. To do this, we study the restriction of to . It turns out that for a suitable class of initial states (roughly speaking, low-energy states) the behaviour of at the vertex does not depend on the initial state, but only on the spectral properties of . The estimates that constrain the behaviour of are expressed in terms of both the parameters , : in concrete situations one can choose to be a suitable function of , to optimize the error. This is particularly simple for free particles, but we prefer to keep the discussion at a general level to prepare the ground for successive generalizations.

### Some definitions

We introduce here the notion of spectrum of the mesoscopic region, and illustrate some results from [grieser08a] that characterize it. Given a generic junction , it is clear that the explicit spectrum cannot be computed in terms of the parameters and , even if some asymptotics may be calculated for specific geometries. Our point of view is that we can formulate some hypotheses on this spectrum regarding its qualitative behaviour near the continuum treshold, that enable us to distinguish between junctions that lead to decoupling or coupling conditions for the limit Quantum Graph.

######
Definition

1 (auxiliary Hamiltonian)

On the mesoscopic region we define the operator

where the Laplace operator is defined by Neumann boundary conditions at and Dirichlet boundary conditions on the rest of . is the auxiliary Hamiltonian on the mesoscopic region. The spectrum of is discrete and it will be denoted by

The spectrum of the auxiliary Hamiltonian is sometimes referred to as the spectrum of the mesoscopic region. This spectrum is analyzed in [grieser08a] in great detail, although in a slighly different setting: the domain is rescaled by a factor and the eigenvalues are multiplied by , accordingly. The dilated mesoscopic region has linear dimension and the branches have tickness ; sometimes one just says that we are working in coordinates such that . In particular, it is shown that any eigenvalue (proper and generalized) of can be approximated by an eigenvalue of and vice-versa. For the point spectrum, the difference between and vanishes exponentially in : in our notation, it is . The eigenfunctions are shown to converge too, in the proper topology, with the same speed.

An immediate consequence of the considerations above is that the first eigenvalues of will converge to the energy levels corresponding to the bound states of ; the other eigenvalues of will “merge into the continuum”.

For each value of , we shall divide the eigenvalues that merge into the continuous spectrum in two classes, according to qualitative properties of the approach to the bottom of the continuous spectrum when and . If there are no elements in the first class, the limit motion on the graph has Dirichlet (decoupling) boundary conditions at the vertex of the graph. The eigenfunctions corresponding to the elements of the first class (we call them resonant sequences) provide “bridges” connecting the different branches of the graph; if there is only one element in this class, the motion on the limit graph is generated by a Laplacian with suitable Kirchhoff-like boundary conditions, which depend on the asymptotic form of this connecting eigenfunction. If there are more than one elements, the problem becomes more complex, and will not be treated here.

Recall that the first eigenvalues of converge to the bound states of the waveguide as ; henceforth, we will loosely refer to all these states as “bound states”, or “localized states”. To proceed further, it will be convenient to introduce the following

######
Definition

2 (spectral gap condition)

The family of (mesoscopic) regions satisfies the spectral gap condition if there exist and such that the eigenvalues of the non-bound states , satisfy, uniformly in

(1) |

We are also interested in waveguides where the gap condition is not satisfied, i.e. there are some non-bound states with eigenvalues that do not satisfy the inequality stated above. We do not treat the general case, but limit ourselves to the following important special case:

######
Definition

3 (resonant sequence condition)

The family of (mesoscopic) regions satisfies the resonant sequence condition if the eigenvalues of the non-bound states , (hence, all of them but the -th) satisfy condition (1) uniformly in , while

(2) |

for some , .

We call the sequence of states which characterize the definition above a resonant sequence of states. The name we have chosen comes from the fact that if the waveguide has a zero-energy resonant sequence (a solution of the Dirichlet problem not in ), then its restriction to provides a resonant sequence. The last statement follows from a variational argument, which bounds the eigenvalue from above, and a result discussed in [grieser08a] which bounds the eigenvalue from below. Note that [grieser08a] proves that in the self-similar case the convergence is exponentially fast. Indeed, including the polynomial bound is useful for generalization to the non-uniform case.

In a self-similar family of waveguides, if a resonance exists for one value of , it exists for all values of . But a resonant sequence may exist even if there is no zero-energy resonance: this will be very useful for generalising the analysis to graphs with many vertices, where self-similarity is lost. We expect the resonant state condition to hold when the waveguide has some resemblance with the domain accessible to conduction electrons of an aromatic molecule^{3}^{3}3The role of a resonant structure has been advocated by B. Pavlov, from a different point of view..

Remark
The inequalities of Equations 1, 2 are written in terms of the parameters , so that their application in the forthcoming Theorems is immediate. But we stress that if we rewrite the Definitions above for the rescaled mesoscopic region of tickness and lenght (“take ”), for which the spectrum is rescaled by a factor , then the eigenvalues are functions of the parameter alone, and Equations 1, 2 become conditions on the asymptotic behaviour of these eigenvalues, for large. Thus, the information about the limit Quantum Graph (wether the junction gives coupling or decoupling gluing conditions) is encoded into the asymptotics of the spectrum of the rescaled mesoscopic region. The parameter will play a role when we discuss the class of states (the low-energy states) that can be approximated by effective wavefunctions on the graph; at that stage, it will be convenient to choose as a function of in order to minimize the error terms.

##### Towards the construction of examples

It would be very interesting to build some concrete examples of geometries such that the Definitions above are fulfilled. We give some suggestions that come from preliminary computations.

We expect that the L-shaped waveguide of Figure 1 falls into Definition 2. This waveguide has one bound state [exner89] (in our notation, ), hence we need to look at the behaviour of the second eigenvalue of the auxiliary Hamiltonian. For this particular geometry it should be possible to compute the asymptotic of the spectrum for large , by adaptating the arguments used in the paper [grieser07]: we expect that in this case the condition of Equation 1 is fulfilled. Note that if is antisymmetric with respect to the reflection through the axis of the waveguide, this follows immediately from a standard bracketing argument.

We also expect that the waveguide depicted in Figure 2 falls into Definition 3 instead, at least for a wise choice of the radii of the junction. The idea is that there are no bound states, so that if we are able to tune the parameters in such a way that a resonance for the infinite waveguide is present, then the resonance induces a resonant sequence by the mechanism described previously. Note that the “hole” is a suggestion that comes from Physics: it mimics a repulsive potential, which is given for example by the core electrons of the carbon atom that sits at the vertex of a graphene layer.

Clearly, it is more difficult to exihibit examples of guides satisfying this hypothesis. It is to be expected that a resonance (or its counterpart, the resonant sequence) is not present for a generic waveguide, accordingly to the general wisdom that decoupling generically occurs for thin Dirichlet waveguides [molchanov07] [molchanov08] [grieser08a].

## 3 Multiscale decoupling

In this section, we want to show that outside the mesoscopic region low-energy wavefunctions are confined to the first transverse mode, and the coefficient solves the free one-dimensional Schrödinger equation; therefore, they can be identified with one-dimensional wavefunctions on . The next sections are devoted to describe the behaviour of the solutions inside the mesoscopic region, that is to say, near the vertex of .

Let belong to the image of the spectral projector associated to . For practical purposes, we may assume that at time zero is approximated by a product state

supported in . This state has finite energy, and is orthogonal to the bound states of : both these properties hold at all times, by unitarity of the time evolution. At any time, the restriction of to admits a convergent expansion in Fourier modes:

where is the -th Dirichlet eigenfunction of on , and is a vector-valued function on ; its components are , the index runs from to and labels the branches of the waveguide. It is useful to think about , for any , as a function on the limiting graph .

We will prove the following

###### Theorem 1 (Multiscale decoupling)

Consider the wavefunction described above, and restrict it to . We can prove that

for some numerical constant . Moreover,

for some numerical constant .

In other words, the components of the wavefunction along the higher transverse modes are suppressed in the limit : the only component which survives is “frozen” in the first transverse mode .

Proof

The first eigenvalues of the auxiliary Hamiltonian converge to the bound states in the sense explained above: the difference is (we will loosely refer to them as “bound states” too); the eigenvectors converge uniformly, with all derivatives, to the respective bound states of restricted to . It follows that if is orthogonal to , its restriction to will be almost orthogonal to , up to an error that is .

In the following we will just say that (restricted to ) is almost orthogonal, or quasi-orthogonal, to , without writing down the exponentially small error: this is because we will obtain estimates that are polynomial in and , so that the exponentially small error is easily reabsorbed; and this will help a lot in keeping the formulas clean.

If the Laplacian on has no eigenvalues below other than the “bound states”, a standard spectral argument implies that

Moreover, by substituting the Fourier expansion of we easily see that

Hence, both these contributions are positive: from the energy bound on we know that their sum is positive and smaller than , and so each one of them (in particular, the integral over ) is bounded by . Notice that if the integral over had been large and negative, the integral over could have been arbitrarily large; this is why we must avoid bound states.

After this simple, but important observation, the first part of the theorem easily follows by substituting the Fourier decomposition of in the integral. We obtain

To complete the theorem, note that . Repeating the previous argument, with simple modifications, one proves that the sum of the norm squared of is bounded: by interpolation, the sum of the squared norms of , for is . This implies that the -norm of , , is : we recall that this implies that the -norm is infinitesimal too, with the same bounds.

Remark Consider the expectation value of an observable, which is either given by the multiplication by a smooth function supported in , or a linear differential operator in the longitudinal derivatives , with coefficients in the smooth functions supported in . Theorem 1 tells us that the expectation values on of these observables converge for small to the expectation values on of the corresponding observables restricted to . Consider for instance a function on (vanishing on ): and say that we want to compute

By Theorem 1, it is easy to see that apart from an error term vanishing as a suitable power of , these expectation values respectively converge to

where is the evaluation of on . Notice that .

The last remark is crucial: it allows us to use instead of to compute the expectation values of important observables, such as the density and (longitudinal) current of particles. Henceforth, we will study the fundamental Fourier component , and systematically neglect the higher-order contributions to , which do not give contribution to the observables of interest.

It is desirable to determine an effective evolution equation for in the limit . Notice that , defined as

is measurable, as its time and space derivatives; the -norms of and of are bounded, hence is in . It is easy to see that satisfies the free (one-dimensional) Schrödinger equation, in weak sense. Indeed, for any smooth test function with support in , and of the form the equations of motion read

and after the integration on the transverse variable the equation reduces to

This equation is the weak form of the free evolution of along the edges of the graph. Since the support of is at a distance not smaller than from the vertex, the equation is satisfied outside the vertex. By a compactness argument, there exists a subsequence having a weak limit , which must satisfy the weak Schrödinger equation on the edges of as well (recall that the initial datum is the same for all values of !). is our limit wavefunction on : once we have proven that the unitary evolution of is uniquely determined, it will be a posteriori clear that any subsequence, and hence the whole sequence converges to the same . Our goal is to use , instead of , to approximate the expectation values of . The next section is devoted to determining the time evolution of , which depends crucially upon the behaviour of at the vertex of .

## 4 The limit motion on the graph

In the following, we will show how one can approximate the low-energy dynamics of the dynamical system with a suitable dynamics on the metric graph . The cornerstone of our approach is the study of the spectrum in the mesoscopic region . The limit dynamics is uniquely identified in this section; more quantitative results (convergence of resolvents and of wave operators) are dealt with in the next sections.

For simplicity, we only treat the case with no external potential. The method can be used also in case there is a continuous potential ; we shall denote by its restriction to the graph. We suggest that such perturbations can be treated by approximating the wave operators on the waveguide with their analogues on : this is sketched in the last section.

### The method of approximating (zero-energy) resonant sequences

We have a sequence of star-shaped waveguides (“-fat” star graphs) which shrink to a star-shaped graph as . The method we describe allows the determination of the boundary conditions at the vertex of , which are needed in order to define the effective dynamics on the graph that, for small , approximates the low-energy dynamics generated by . These boundary conditions depend on the shape of in the neighborhood of the vertex region . The dependence is through the spectra of the auxiliary operators acting on the mesoscopic region .

We will see in the next paragraph that the control of the effective wavefunction on the limit graph near the vertex is possible if we consider a particular class of initial states (low-energy states) and choose wisely the parameters , . But once we restrict to this class of initial states, the gluing conditions at the vertex of the graph depend on the geometry of the waveguide and nothing else: the dependence is through the asymptotics (in the parameter ) of the eigenvalues of the rescaled mesoscopic region.

### The wavefunctions near the vertex

Let , smooth, normalized, energy-bounded state orthogonal to the bound states of . Consider the finite cylinders

contained in the mesoscopic region: is part of the boundary of the mesoscopic region. Notice that (up to a zero-measure boundary)

The function can be Fourier decomposed as usual:

We can prove the following

###### Theorem 2

Assume that the spectral gap condition holds. If belongs to , then the -norm of

is bounded by , where , are numerical constants.

An important consequence of this theorem is the following. Let be the weak limit of any convergent subsequence . It is in , therefore each component , is right-continuous at the vertex . Since its approximants are uniformly bounded near the vertex by an infinitesimal quantity, it follows that

Proof

is a state with energy bounded by . Application of the spectral condition, together with (quasi)-orthogonality to the negative-energy states of

^{4}^{4}4we recall that these states approximate the bound states of up to an exponentially small error, see the Definition of . We omit the exponentially small errors from the formulas, to make them easier to read., tells us that

This inequality can be written in a more useful form:

Now, we recall that

(this is easily seen by expanding in Fourier series) and so

from which we deduce

This states that the norm of (and hence of ) is bounded by a suitable power of (not uniformly in ).

For any smooth function on the interval and any point , the following Poincaré-like estimate holds:

Consider the restriction of to : we write

The remainder is the sum of the components of along the higher transverse modes. From the Poincaré estimate applied to ,

Since

is finite, we obtain an estimate for in terms of , :

Using this last equation can be rewritten as:

If , the term in curly brackets is bounded by as , irrespective of the particular choice of the parameter that we have introduced. If then we need to restrict the choice of , in order to bound with an infinitesimal quantity. In fact, by choosing for sufficiently close to , the last equation becomes

and the right hand side clearly vanishes in the limit for any value of , if is suitably chosen.

In the following it will be useful to know something about the restriction of to the boundary of :

###### Theorem 3

Assume that the spectral gap condition holds. If belongs to , then the -norm of

is bounded by , where , are numerical constants. The constants are not necessarily the same of theorem 2.

Proof

Consider the sum of the higher modes : using in the Poincaré-like estimate and summing up from to , we get

(all norms are -norms). We use the bounds obtained in Theorem 1 for the sum of the higher modes: the proof then mimics the proof of theorem 2.

Another simple modification of Theorem 2 allows to state that, if the resonant state condition holds:

###### Theorem 4

Assume that the resonant sequence condition holds. If belongs to , define

and

Then, the norm of , restricted to the segment , is bounded by , where , are numerical constants. Recall that is the resonant sequence of .

Pictorially, we may say that “relaxes” to .

### Dirichlet dynamics on graphs

In the spectral gap case, our argument is already sufficient to establish that the limit dynamics on the star graph is the decoupled one with Dirichlet boundary conditions at the vertex. Let be a solution of the Schrödinger equation in . We know that

To our purposes, can be neglected. The function has a weak limit , for ; the limit exists by compactness (choose a subsequence), and it satisfies the weak Schrödinger equation on . By Theorem 2, satisfies Dirichlet boundary conditions at the vertex :

and this fixes the unitary evolution of on the limit graph completely.

Since any convergent subsequence has the same limit at time zero, and the limit evolves with the same equation, then the whole family converges to the same limit , at all times.

### Kirchhoff dynamics on graphs

We have seen that the eigenstates of which satisfy Equation 1 (the spectral gap condition) give in the limit a contribution to the limit wavefunction which is negligible near the vertex. The limit behaviour at the vertex other than Dirichlet conditions is entirely due to the resonant sequence of eigenstates; we must therefore analyze in detail this resonant sequence. In particular, we are interested in finding how the resonant sequence determines the parameters which characterize the self-adjoint extension of the Laplacian on the graph.

#### The resonant wavefunction

Let the complex numbers be defined by

where the term stands for the sum of the higher modes , . We will choose the normalization of so that

With this normalization the numbers depend on and only through the ratio : this is a consequence of the homotheticity of the family of waveguides . Define

(we assume that the limit exists). Note that by continuity

Consider the resonant eigenvalue . Set ; , so that in our setting . By a simple scaling argument,

where is the -th eigenvalue of the rescaled mesoscopic region of thickness and length . Define

(we assume that this limit exists, too). is (minus) the derivative of the resonant eigenvalue with respect to the parameter . We point out that is nonnegative.

The following estimates are useful to recover the boundary conditions at the vertex. Consider the function defined on , solution of the inhomogeneous Dirichlet problem

###### Theorem 5

The following formula holds true:

Proof

Observe that is the solution to an inhomogeneous problem similar to the one solved by , but with different boundary data on :

Since both the boundary data and the eigenvalues of the two systems converge, we expect the solutions to converge to each other in the limit in a suitable Sobolev topology. If the boundary data converge in , converges to uniformly, but their derivatives (in particular, the normal derivatives at the boundary) in general do not.

Consider the Gauss-Green identity

and apply it for , , ; using the equations

we get

which is equal to

Note that

and

Thus

(note that , this term was introduced to emphasize the fact that the definition of involves “derivative” of the resonant energy level .

#### Kirchhoff graphs

The argument runs exactly in the same way as in the Dirichlet case, except for establishing the vertex conditions, which is what we are going to do. Let us consider a generic wavefunction in ; recall that

by evaluating this expression on we see that the right hand side equals

(dots stand for the negligible higher modes). By Theorem 4 we see that

and since

the comparison of the coefficients of the fundamental mode gives

which, for the limit wavefunction , reads

Now consider the restriction of to the mesoscopic region . We make use of the Gauss-Green identity:

substituting for ,