Abstract
We construct models of manyparticle quantum graphs with singular twoparticle contact interactions, which can be either hardcore or interactions. Selfadjoint realisations of the twoparticle Laplacian including such interactions are obtained via their associated quadratic forms. We prove discreteness of spectra as well as Weyl laws for the asymptotic eigenvalue counts. These constructions are first performed for two distinguishable particles and then for two identicle bosons. Furthermore, we extend the models to bosons with twoparticle interactions, thus implementing the LiebLiniger model on a graph.
Quantum graphs with twoparticle
[5mm] contact interactions
Jens Bolte
Department of Mathematics
Royal Holloway, University of London
Egham, TW20 0EX
United Kingdom
1 Introduction
This paper is the second in a serious of papers developing models of interacting, nonrelativisitc manyparticle systems on (compact) graphs. The underlying oneparticle quantum graphs are well established models for a variety of problems in quantum mechanics. Interest in such oneparticle models surged after Kottos and Smilansky [KS99b] had discovered that the eigenvalues of quantum Hamiltonians describing single particles on graphs possess the same correlations as eigenvalues of random hermitian matrices. Henceforth, quantum graphs proved to be very successful models in the area of quantum chaos [GS06] and beyond [EKK08].
In our first paper [BK11] we introduced systems of two particles on compact metric graphs. We identified selfadjoint realisations of the twoparticle Laplacian that describe singular twoparticle interactions that are located in the vertices of the graph. In that context we analysed the Laplacians indirectly, by first constructing closed, semibounded quadratic forms and then identifying the selfadjoint operators that are uniquely associated with the quadratic forms. Using quadratic forms allowed us, furthermore, to prove that the Laplacians have compact resolvents and thus possess purely discrete spectra. Moreover, with a bracketing argument we were able to prove a Weyl law for the asymptotic eigenvalue count.
The goal of this paper now is to introduce twoparticle interactions of a different kind, modelling shortrange interactions in terms of singular contact interactions that are formally given by a Hamiltonian
(1.1) 
Here and are coordinates for the positions of the two particles on the edges of the graph and is a coupling parameter, such that corresponds to repulsive interactions. Contact interactions of this kind play a prominent role in the description of BoseEinstein condensates (BEC), leading to GrossPitaevskii equations, in plasma physics and in solid state physics (see, e.g., [LSSY05, CCG11]). They are also of interest from a mathematical point of view: models of (an arbitrary number of) particles on the real line with Hamiltonian (1.1) are completely solvable in the sense that the eigenfunctions can be constructed explicitly, see [Yan67, AFK02, AFK04]. As in the oneparticle case the connectivity of a graph, however, adds sufficient complexity to the problem to render it not solvable in that sense. Therefore, bosonic manyparticle systems on graphs are expected to show generic manyparticle properties of systems confined to one spatial dimension.
In this paper we shall follow our previous approach [BK11] in that we first introduce suitable quadratic forms, show that they are closed and semibounded, and then identify the corresponding selfadjoint operators. These operators shall always be twoparticle Lapacians, however, with suitable domains that implement singular twoparticle contact interactions. The resulting operators are then rigorous versions of the formal Hamiltonian (1.1). In order to achieve this the domains of the operators are required to contain jump conditions along diagonals for derivatives of the functions in the domain. A similar construction was already given by Harmer [Har07, Har08], who considered two particles on a star graph (with semiinfinite edges). Our construction works for any compact metric graph and allows for straightforward generalisations to somewhat broader classes of singular contact interactions of which the type interactions (1.1) form a prominent subclass.
The paper is organised as follows: In Section 2 we review important properties of oneparticle quantum graphs and introduce relevant concepts and notations that we are using in the following sections. Section 3 is then devoted to the construction of the contact interactions via quadratic forms for two distinguishable particles on a general compact, metric graph. In that context we encounter the problem of elliptic regularity [Dob05] in the same way as previously [BK11], suggesting an analogous solution for a similar class of boundary conditions. In Section 4 we first implement a bosonic realisation of a particle exchange symmetry and hence obtain a rigorous construction of the bosonic version of the operator (1.1). We then extend this construction to bosons on a general compact graph, for which the equivalent to the formal operator (1.1) is
(1.2) 
Here are the particle positions and is the Laplacian in these variables. Hence, the model we are investigating is an extension to quantum graphs of the wellknown LiebLiniger model [LL63] that has been studied extensively in the context of BEC.
2 Preliminaries
Before introducing manyparticle systems on graphs we briefly describe oneparticle quantum graphs. They form the basis for the tensorproduct construction of manyparticle quantum systems on graphs.
The classical configuration space of a quantum graph is a compact metric graph, i.e., a finite graph with vertices and edges . The latter are identified with intervals , , thus introducing a metric on the graph, see [KS99b, KS99a, Kuc04, GS06] for details.
Functions on the graph are collections of functions on the edges,
(2.1) 
so that spaces of functions on are (finite) direct sums of the respective spaces of functions on the edges. The most relevant space is the oneparticle Hilbert space
(2.2) 
and all other spaces are constructed in a similar way.
The oneparticle Hamiltonian is a Laplacian, acting as a secondorder differentiation,
(2.3) 
on . We here use the index to indicate that this is a oneparticle Laplacian.
Domains of selfadjoint realisations of the Laplacian are characterised in terms of boundary conditions in the vertices. These require boundary values
(2.4) 
of functions and (inward) derivatives,
(2.5) 
as well as projectors and acting on the space of boundary values, and selfadjoint endomorphisms of .
The selfadjoint realisations of can be identified via the quadratic forms that are uniquely associated with them [Kuc04].
Theorem 2.1 (Kuchment).
The quadratic form
(2.6) 
with domain
(2.7) 
is closed and bounded from below. The unique, selfadjoint and semibounded operator associated with this form is the oneparticle Laplacian with domain
(2.8) 
A twoparticle quantum system requires the tensor product of two oneparticle Hilbert spaces,
(2.9) 
For a quantum graph this means that
(2.10) 
such that vectors are collections of functions defined on the rectangles . Their disjoint union is denoted as
(2.11) 
so that one may view as
(2.12) 
We shall use a similar notation for other function spaces.
As a differential operator, the twoparticle Laplacian acts as
(2.13) 
and hence has the same form as a Laplacian in . Defined on the domain , this operator is symmetric, but not selfadjoint.
Selfadjoint realisations of the twoparticle Laplacians can either represent noninteracting particles, or introduce twoparticle interactions via boundary conditions. A particular class of singular twoparticle interactions that are localised in the vertices was established in [BK11]. Here we shall introduce twoparticle contact interactions that are localised on the edges.
3 Contact interactions on a general compact graph
The interactions we have in mind are intended to model two pointlike particles on a graph that interact when they hit each other, i.e., when they are located in the same position. This requires, in particular, that they are on the same edge. Hence, the subset of the twoparticle configuration space (2.11) where these interactions take place consists of the diagonals of the squares . Singular interactions require a dissection of the configuration space along these subspaces, and suitable matching conditions for functions and their derivatives along the boundaries introduced by the dissection. We therefore define the ‘dissected’ configuration space
(3.1) 
where and . Functions on are denoted as . The components for are defined on , whereas with defined on .
The twoparticle Hilbert space (2.10) can then also be viewed as
(3.2) 
Boundary values of functions are encoded in vectors
(3.3) 
We distinguish components with from those with , as in the latter case additional boundary values along diagonals have to be taken into account. More specifically, when we define
(3.4) 
where . When boundary values along the diagonals of the squares have to be added, including those for derivatives. Noting that the inward normal derivatives along the ‘diagonal’ part of the boundary of are
(3.5) 
we set
(3.6) 
for . Altogether, the vectors (3.3) of boundary values have components.
As a next step we introduce the bounded and measurable maps that are required to fulfil

is an orthogonal projector,

is a selfadjoint endomorphism on ,
for a.e. . We then introduce two bounded and selfadjoint operators, and , on . They are defined to act as and on .
Our aim is to obtain selfadjoint realisations of the twoparticle Laplacian , see (2.13), that represent twoparticle contact interactions and are extensions of defined on the domain . In analogy to the oneparticle case (2.8), as well as the case of singular interactions covered in [BK11], their domains should be given in the form
(3.7) 
In order to clearly distinguish the boundary conditions that induce contact interactions from other kinds of boundary conditions we rearrange the order of terms in the boundary vectors. We first list, for each edge , the lower two boundary values in (3.6), and then, for each pair , either the four components in (3.4) or the upper four components of (3.6). That way one achieves a decomposition of the space of boundary values according to
(3.8) 
Here , with , contains the boundary values (3.6) along diagonals, whereas , with , contains the remaining boundary values (3.4) and (3.6), which are associated with vertices. A separation of contact interactions from any other boundary effects requires to choose and as blockdiagonal with respect to the decomposition (3.8). From now on we assume this to be the case.
For the restriction of and to we assume the same conditions as in [BK11]. For most purposes, however, it is sufficient to suppose that there are no twoparticle interactions in the vertices. In [BK11] the noninteracting boundary conditions were characterised as follows: The restrictions of and to are independent of and blockdiagonal with respect to a decomposition of according to the index in (3.3).
The restriction of and to should, first of all, be blockdiagonal with respect to a decomposition of that space according to the edges in order to avoid ‘contact’ interactions across edges. Further restriction are not necessary, but we identify the following two cases as of particular interest because they correspond to a Hamiltonian of the form (1.1).
Definition 3.1.
Let be Lipschitz continuous. Then a contact interaction is said to be of

type with (variable) strength , if is continuous across diagonals,
(3.9) and satisfies jump conditions for the normal derivatives,
(3.10) 
hardcore type, if it satisfies Dirichlet boundary conditions along diagonals.
We remark that contact interactions of the type can be seen as a rigorous realisation of a Hamiltonian
(3.11) 
The case for all corresponds to repulsive interactions and is the most relevant case for models of actual particles on a graph. Hardcore interactions follow from such a formal Hamiltonian in the limit .
Following our intention to represent domains of twoparticle Laplacians in the form (3.7) we have to choose the maps and in such a way that their restrictions to the edge subspace of are
(3.12) 
and
(3.13) 
in order to generate type contact interactions. Hardcore interactions require the choice and .
Our approach to selfadjoint realisations of the Laplacian uses suitable quadratic forms, which are uniquely associated with these operators.
Proposition 3.2.
Assume that the maps are bounded and measurable. Then the quadratic form
(3.14) 
with domain
(3.15) 
is closed and semibounded.
Proof.
The proof follows by using the same steps as in the corresponding proof in [BK11]. The only consideration that has to be added concerns the upper bound
(3.16) 
To estimate the righthand side one requires the bound,
(3.17) 
to hold for all , where . The contribution from the rectangles (with ) in the decomposition (3.1) can be dealt with as in [BK11] and is based on a result in [Kuc04]. For the triangles we note that close to the corners with angles this method fails. However, one can always reflect functions across edges, define them on suitable squares and then apply the bound as before for the rectangles. The proof then continues as in [BK11]. ∎
According to the representation theorem for quadratic forms (see, e.g., [Kat66]) there exists a unique selfadjoint and semibounded operator with domain that is associated with the quadratic form . It is not immediately clear, however, that the functions in possess regularity. If this is the case we say, for short, that the quadratic form is regular. We note that a selfadjoint realisations of with domain (3.7) would correspond to a regular form.
Under an additional (mild) assumption a regular quadratic form indeed leads to a twoparticle Laplacian with domain (3.7).
Proposition 3.3.
Suppose that the map is of class and that the quadratic form with domain is regular. Then the unique, selfadjoint and semibounded operator that is associated with this form is the twoparticle Laplacian with domain .
Proof.
The proof can essentially be taken over verbatim from the corresponding proof in [BK11]. It is based on the representation theorem for quadratic forms, which implies that for each there exists a unique such that
(3.18) 
When , an integration by parts of (3.18) implies that acts as a twoparticle Laplacian . In the general case of a the integration by parts yields an additional boundary term,
(3.19) 
that is required to vanish. Following Lemma 3.13 in [BK11], which has an immediate generalisation to the present case, the set is dense in . Hence, , or
(3.20) 
This condition finally implies that . ∎
As mentioned above, the quadratic forms in Proposition 3.2 are not necessarily regular. Since our focus is on contact interactions of  or hardcoretype, it is sufficient to consider these cases. However, as in [BK11] we have to add one additional assumption on the restrictions of the projectors to . Splitting into the two subspaces spanned by the upper two and the lower two components of (3.4) as well as the upper two and middle two components of (3.6), respectively, we require to be blockdiagonal with respect to this decomposition.
This then leads to the main result of this section.
Theorem 3.4.
In addition to the assumption made for the maps and above, suppose that is of class and is Lipschitz continuous. Furthermore, for with some assume that the restriction of to is diagonal with diagonal entries zero or one and, in the case of type interactions, that for those . Then the quadratic form is regular.
Proof.
First note that it is enough to show regularity near the corners of adjacent to the diagonal. The regularity away from the diagonal of as well as regularity in the rectangles with was already established in [BK11]. In addition, the regularity along the diagonals in the interior of can be readily established using the same methods as in [BK11].
The assumptions made on imply that on the edges of the squares the functions in satisfy either Dirichlet or Neumann boundary conditions near the corners. Along diagonals we consider the projections
(3.21) 
The goal is to show that, close to the corners, both and are of class . For that purpose one introduces suitable cutoffs that restrict the functions (3.21) to neighbourhoods of the corners. This eventually implies that .
We recall the conditions (3.10) which imply that
(3.22) 
on the diagonal. Hence, satisfies (variable) Robin boundary conditions on the diagonal. By construction, vanishes on the diagonal so that near the corners of adjacent to the diagonal, where is supposed to be constant, satisfies a combination of Dirichlet, Neumann or standard Robinboundary conditions. In all such cases regularity is well known to hold [Dau88, Gri85]. ∎
One naturally expects the twoparticle operators representing contact interactions to possess purely discrete spectra of the form (i.e., eigenvalues are counted with their multiplicities and do not accumulate at any finite value). Moreover, the asymptotic distribution of eigenvalues, as given by the asymptotic behaviour of the eigenvaluecounting function
(3.23) 
should follow a Weyl law. We shall now prove a Weyl law for repulsive contact interactions. This includes hardcore and interactions with . The general requirement is that is negative definite (compare (3.13)).
Proposition 3.5.
Let be a selfadjoint realisation of the twoparticle Laplacian with repulsive contact interaction as described in Proposition 3.3. Then this operator has compact resolvent. In particular, its spectrum is purely discrete and only accumulates at infinity. Furthermore, the counting function (3.23) obeys the Weyl law
(3.24) 
where is the total length of the graph.
Proof.
The Hilbert space is compactly embedded in . Since the form norm is equivalent to the norm, the Hilbert space is also compactly embedded in . Hence the operator associated with the quadratic form has compact resolvent [Dob05].
The Weyl law follows from a standard bracketing argument [RS78] based on a comparison with two suitable operators (quadratic forms), see also [BE09, BK11].
The first operator, , is the DirichletLaplacian, and is characterised by the projector as well as . Given an operator the second comparison operator, , is the Laplacian given by the projector as well as
(3.25) 
where . The associated quadratic forms therefore satisfy the following inclusions of their domains,
(3.26) 
Hence [RS78], it follows that the related eigenvaluecounting functions satisfy
(3.27) 
As both and satisfy the Weyl law (3.24), the same asymptotics hold for . ∎
4 Contact interactions for bosons
So far we assumed to have two nonidentical particles on a graph. We now implement an exchange symmetry for two identical bosonic particles. Their states are described in the symmetric twoparticle Hilbertspace . The orthogonal projection from to the bosonic subspace
(4.1) 
acts on components of with as
(4.2) 
whereas on the components with its action reads
(4.3) 
Due to this symmetry it would be sufficient to keep only components with in addition to the diagonal components with . For simplicity, when comparing to the previous section we, however, keep all components. We then denote the images of function spaces under the projection to their bosonic subspaces as, e.g.,
(4.4) 
We note that whenever is in , the underlying symmetry implies the relations
(4.5) 
when , as well as
(4.6) 
Due to the bosonic symmetry it is possible to reduce the number of components in the vectors of boundary values (3.3). When , it suffices to keep the upper two components in each of the vectors (3.4), whereas for we use
(4.7) 
with . The space of boundary values therefore has dimension , and decomposes in analogy to (3.8).
We also need the bounded and measurable maps , where

is an orthogonal projector,

is a selfadjoint endomorphism on ,
for a.e. . The space of boundary values decomposes in the same way as (3.8), however, the edge subspaces are now onedimensional. This forces the equivalent of (3.12) to be a projector on and to take values
(4.8) 
Here corresponds to a Dirichlet condition in the point along the diagonal, whereas imposes no condition. Hence, when type interactions are considered we choose , and in the case of hardcoreinteractions is chosen. Likewise, the equivalent of (3.13) is
(4.9) 
for interactions, and for interactions of hardcore type.
We can now set up the following quadratic form,
(4.10) 
with domain
(4.11) 
As this is the restriction of a quadratic form on to , all results of Section 3 carry over: Propositions 3.2 and 3.3 imply that the quadratic form is closed and semibounded; when is of class and the form is regular, the associated selfadjoint operator is the bosonic twoparticle Laplacian with domain
(4.12) 
According to Theorem 3.4, when the consitions of that theorem are fulfiled the quadratic forms leading to type and hardcoreinteractions are regular.
We remark that for interactions one can use the decomposition (3.8) of the space of boundary values and the explicit expression (4.9) to rewrite the quadratic form as
(4.13) 
In the same way, the form domain takes the form
(4.14) 
Due to the bosonic projection , which commutes with any of the twoparticle Laplacians, asymptotically half of the spectrum of a Laplacian is projected to the bosonic Hilbert space , so that Proposition 3.5 implies the Weyl law
(4.15) 
for the asymptotics of the eigenvalue count restricted to .
Our goal now is to study bosonic manyparticle systems on graphs. Eventually, these have to be described in the bosonic Fock space over the oneparticle Hilbert space. Since it suffices, however, to consider each particle space separately, we here only consider a fixed particle number . In that context we shall introduce twoparticle interactions that are (formally) of the type,
(4.16) 
Due to the bosonic symmetry, on suitable functions the quadratic form associated with such an operator will be