We construct models of many-particle quantum graphs with singular two-particle contact interactions, which can be either hardcore- or -interactions. Self-adjoint realisations of the two-particle 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 two-particle interactions, thus implementing the Lieb-Liniger model on a graph.
Quantum graphs with two-particle
[5mm] contact interactions
Department of Mathematics
Royal Holloway, University of London
Egham, TW20 0EX
This paper is the second in a serious of papers developing models of interacting, non-relativisitc many-particle systems on (compact) graphs. The underlying one-particle quantum graphs are well established models for a variety of problems in quantum mechanics. Interest in such one-particle 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 self-adjoint realisations of the two-particle Laplacian that describe singular two-particle interactions that are located in the vertices of the graph. In that context we analysed the Laplacians indirectly, by first constructing closed, semi-bounded quadratic forms and then identifying the self-adjoint 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 two-particle interactions of a different kind, modelling short-range interactions in terms of singular contact interactions that are formally given by a Hamiltonian
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 Bose-Einstein condensates (BEC), leading to Gross-Pitaevskii 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 one-particle case the connectivity of a graph, however, adds sufficient complexity to the problem to render it not solvable in that sense. Therefore, bosonic many-particle systems on graphs are expected to show generic many-particle 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 semi-bounded, and then identify the corresponding self-adjoint operators. These operators shall always be two-particle Lapacians, however, with suitable domains that implement singular two-particle 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 semi-infinite edges). Our construction works for any compact metric graph and allows for straight-forward 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 one-particle 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
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 well-known Lieb-Liniger model [LL63] that has been studied extensively in the context of BEC.
Before introducing many-particle systems on graphs we briefly describe one-particle quantum graphs. They form the basis for the tensor-product construction of many-particle 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,
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 one-particle Hilbert space
and all other spaces are constructed in a similar way.
The one-particle Hamiltonian is a Laplacian, acting as a second-order differentiation,
on . We here use the index to indicate that this is a one-particle Laplacian.
Domains of self-adjoint realisations of the Laplacian are characterised in terms of boundary conditions in the vertices. These require boundary values
of functions and (inward) derivatives,
as well as projectors and acting on the space of boundary values, and self-adjoint endomorphisms of .
The self-adjoint realisations of can be identified via the quadratic forms that are uniquely associated with them [Kuc04].
Theorem 2.1 (Kuchment).
The quadratic form
is closed and bounded from below. The unique, self-adjoint and semi-bounded operator associated with this form is the one-particle Laplacian with domain
A two-particle quantum system requires the tensor product of two one-particle Hilbert spaces,
For a quantum graph this means that
such that vectors are collections of functions defined on the rectangles . Their disjoint union is denoted as
so that one may view as
We shall use a similar notation for other function spaces.
As a differential operator, the two-particle Laplacian acts as
and hence has the same form as a Laplacian in . Defined on the domain , this operator is symmetric, but not self-adjoint.
Self-adjoint realisations of the two-particle Laplacians can either represent non-interacting particles, or introduce two-particle interactions via boundary conditions. A particular class of singular two-particle interactions that are localised in the vertices was established in [BK11]. Here we shall introduce two-particle 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 point-like 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 two-particle 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
where and . Functions on are denoted as . The components for are defined on , whereas with defined on .
The two-particle Hilbert space (2.10) can then also be viewed as
Boundary values of functions are encoded in vectors
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
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
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 self-adjoint endomorphism on ,
for a.e. . We then introduce two bounded and self-adjoint operators, and , on . They are defined to act as and on .
Our aim is to obtain self-adjoint realisations of the two-particle Laplacian , see (2.13), that represent two-particle contact interactions and are extensions of defined on the domain . In analogy to the one-particle case (2.8), as well as the case of singular interactions covered in [BK11], their domains should be given in the form
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
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 block-diagonal 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 two-particle interactions in the vertices. In [BK11] the non-interacting boundary conditions were characterised as follows: The restrictions of and to are independent of and block-diagonal with respect to a decomposition of according to the index in (3.3).
The restriction of and to should, first of all, be block-diagonal 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).
Let be Lipschitz continuous. Then a contact interaction is said to be of
-type with (variable) strength , if is continuous across diagonals,
and satisfies jump conditions for the normal derivatives,
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
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 two-particle 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
in order to generate -type contact interactions. Hardcore interactions require the choice and .
Our approach to self-adjoint realisations of the Laplacian uses suitable quadratic forms, which are uniquely associated with these operators.
Assume that the maps are bounded and measurable. Then the quadratic form
is closed and semi-bounded.
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
To estimate the right-hand side one requires the bound,
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 self-adjoint 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 self-adjoint 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 two-particle Laplacian with domain (3.7).
Suppose that the map is of class and that the quadratic form with domain is regular. Then the unique, self-adjoint and semibounded operator that is associated with this form is the two-particle Laplacian with domain .
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
When , an integration by parts of (3.18) implies that acts as a two-particle Laplacian . In the general case of a the integration by parts yields an additional boundary term,
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
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 hardcore-type, 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 block-diagonal with respect to this decomposition.
This then leads to the main result of this section.
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.
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
The goal is to show that, close to the corners, both and are of class . For that purpose one introduces suitable cut-offs that restrict the functions (3.21) to neighbourhoods of the corners. This eventually implies that .
We recall the conditions (3.10) which imply that
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 Robin-boundary conditions. In all such cases regularity is well known to hold [Dau88, Gri85]. ∎
One naturally expects the two-particle 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 eigenvalue-counting function
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)).
Let be a self-adjoint realisation of the two-particle 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
where is the total length of the graph.
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 first operator, , is the Dirichlet-Laplacian, 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
where . The associated quadratic forms therefore satisfy the following inclusions of their domains,
Hence [RS78], it follows that the related eigenvalue-counting functions satisfy
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 non-identical particles on a graph. We now implement an exchange symmetry for two identical bosonic particles. Their states are described in the symmetric two-particle Hilbert-space . The orthogonal projection from to the bosonic subspace
acts on components of with as
whereas on the components with its action reads
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.,
We note that whenever is in , the underlying symmetry implies the relations
when , as well as
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
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 self-adjoint endomorphism on ,
for a.e. . The space of boundary values decomposes in the same way as (3.8), however, the edge- subspaces are now one-dimensional. This forces the equivalent of (3.12) to be a projector on and to take values
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 hardcore-interactions is chosen. Likewise, the equivalent of (3.13) is
for -interactions, and for interactions of hardcore type.
We can now set up the following quadratic form,
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 semi-bounded; when is of class and the form is regular, the associated self-adjoint operator is the bosonic two-particle Laplacian with domain
According to Theorem 3.4, when the consitions of that theorem are fulfiled the quadratic forms leading to -type and hardcore-interactions are regular.
In the same way, the form domain takes the form
Due to the bosonic projection , which commutes with any of the two-particle 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
for the asymptotics of the eigenvalue count restricted to .
Our goal now is to study bosonic many-particle systems on graphs. Eventually, these have to be described in the bosonic Fock space over the one-particle 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 two-particle interactions that are (formally) of the type,
Due to the bosonic symmetry, on suitable functions the quadratic form associated with such an operator will be