Contact interactions: the two-dimensional case

We prove that in two dimensions the contact interaction of two wave functions is represented by a self-adjoint operator. Together with a confining potential, this provides a stable condensate of two-particle systems. Recall that in three dimensions one has a condensate of four wave functions in contact interaction (the Bose–Einstein condensate).


Introduction
In [1,2] we introduced contact interaction in dimension 3; there are two types of interaction: weak and strong . Weak contact is related to superfluidity and is associated to a zero energy resonance. Strong contact is connected to a Bose-Einstein condensate, a collection of clusters of two pairs of wave functions in strong contact. The particle in the cluster can be bosons or spin 1 2 fermions. Weak and strong interactions are different self-adjoint extensions of the symmetric operator H 0 0 , the free Hamiltonian H 0 restricted to functions that vanish on a neighbourhood of the "contact manifold" ≡ x i = x j , i = j.
Strong contact is obtained by requiring that the domain of the extension contains functions that have a cups at . For weak contact, the domain contains functions that take a negative value c at . In three dimensions, this implies the presence of a zero energy resonance.
In R 3 contact interactions are limits, in strong resolvent convergence sense, of Hamiltonians with potentials that scale as V = 1 k V ( |x 1 −x 2 | ) where k = 3 for strong contact and k = 2 for weak contact.
Here, we consider the two-dimensional case. We will prove that the outcome is completely different.
In L 2 (R 2 ) there is only one type of contact interaction. It is a self-adjoint extension of the symmetric operator defined by the two-body free Hamiltonian on functions that vanish in a neighbourhood of the contact manifold ≡ The extended domain contains functions that take a finite value c at contact. We shall prove that in two dimensions the Hamiltonian that describes contact interaction is limit in strong resolvent sense of Hamiltonians with negative potentials that scale as V =

Contact interactions as self-adjoint extensions: the case of two dimensions
As in [1,2] we construct contact interactions as self-adjoint extensions of the symmetric operator H 0 0 , the free Hamiltonian restricted to functions that vanish in a neighbourhood of the contact manifold .
The extension is obtained by requiring that the functions in the extended domain take at the value c c; c negative means attraction. . We will prove that the Hamiltonian of contact interaction is the limit in strong resolvent sense of Hamiltonians with potentials V = C 2 V ( |x 1 −x 2 | ) C < 0. The parameter C is a linear function of the parameter c. As in [1,2] for the proof, we make use of a map (the Krein K) from L 2 (R 4 ) to a space of less regular functions, the Minlos space [3] M. By duality the system is more regular in M.
Let H 0 0 be the symmetric operator defined as the free Hamiltonian H 0 = − 1 − 2 restricted to functions that have support away from ; the operator H 0 0 is symmetric but not self-adjoint.
Let (x 1 − x 2 ) be the distribution supported by that is the weak limit of the potentials V when → 0.

The map K acts as
Notice that the distribution formally commutes with the free Hamiltonian. (This is easily seen in Fourier space.) In two dimensions, the degree of the Hamiltonian as an elliptic operator and the dimension of the space (and therefore the dimension of the region of the contact) are the same.
As in the case of strong contact in three dimensions, in M the potential and kinetic terms are self-adjoint operators; they have opposite signs and the same singularity (in relative coordinates ) on functions that are supported at contact. Therefore, [4,5] they define in M a well-ordered one-parameter family of self-adjoint operators.
Inverting the map, due to the change of metric topology one has a well-ordered sequence of weakly closed quadratic forms.
Gamma convergence [6] selects the infimum that is strictly convex. (The perturbation acts only on the s-waves.) The Gamma limit F(y) is the quadratic form defined by the relations The first condition implies that F(y) is a common lower bound for the function F n , and the second implies that this lower bound is optimal.
The condition for the existence of the Gamma limit is that the sequence be contained in a compact set for the topology of Y ( so that a Palais-Smale convergent sequence exists).
In our case, Gamma convergence holds because there is no zero energy resonance, and therefore, the sequence of quadratic forms lies in a compact domain in a Sobolev topology.
Since the interaction is invariant under rotations, the limit form is strictly convex, and therefore, by a theorem of Kato [7] it can be closed strongly and gives a self-adjoint operator, the Hamiltonian of the system. We denote it by K .
The potential term is strictly negative, and therefore, the approximating -Hamiltonians form a decreasing sequence and any of them, as quadratic form, is larger than any of the quadratic form in the sequence of quadratic forms considered in the Gamma convergence, since they all correspond to the limit → 0.
By monotonicity the resolvent set R(K ) of K (the complement of the spectrum) is smaller than the resolvent set R of any of the H .
Gamma convergence implies strong resolvent convergence.
Since the limit of the quadratic forms exists and is unique, also the sequence of the resolvents of the approximating Hamiltonians converges strongly to the resolvent of the Hamiltonian K .
The convergence is in the strong resolvent sense; since R(K ) ⊂ R , the limit is the resolvent of the self-adjoint operator K and the convergence is in norm.
We have proved.

Proposition In two dimensions the Hamiltonian of contact of two particles is a self-adjoint operator which is the limit, in strong resolvent sense, of Hamiltonians H
, c < 0 There are no operator convergence and no estimate of the rate of convergence.
Remark 1 Since the two-particle system is invariant under translations and also invariant under rotation in the centre of mass reference frame, the system can be described in polar coordinates in the centre of mass frame.
One can verify that the radial part of the system is the one-dimensional Laplacian on R + with Neumann boundary conditions at the origin and the constant c is related to the value of the wave function at the origin.
Remark 2 Norm resolvent convergence of H when → 0 is proved in [8] for two wave functions through a very long chain of hard estimates; no description is given in the limit operator.
Norm convergence of the resolvents implies convergence to the resolvent of a limit selfadjoint operator. (Resolvent identities are satisfied also for the limit Hamiltonian.) This operator is the Hamiltonian of contact interaction of two wave functions in R 2 .
The result of [8] leads therefore support to our mathematical procedure. The use of Gamma convergence makes the proof of convergence much easier.

Clusters on N wave functions
In two dimensions two particles in contact define a stable system, i.e. a system described by a self-adjoint Hamiltonian.
Recall that by contact interaction we always mean that the interaction is attractive, i.e. limit of short-range negative potentials.
One can now add one particle in contact with either of the two. This gives a stable system of three particles in contact described by a self-adjoint Hamiltonian.
As for the case of two wave functions, this operator is obtained through a Krein map, defined as in the case of two particles.
In the space M, one has a family of self-adjoint operators.
Since the interactions are independent, one has again a one-parameter family of self-adjoint operator.
Again coming back to L 2 (R 6 ), one has an ordered family of quadratic forms and Gamma convergence selects the infimum that can be closed strongly and gives a self-adjoint operator, the Hamiltonian of the system.
Since the potentials V are strictly negative, the sequence of quadratic forms for the contact interaction is strictly smaller than the forms of the three-body where H 0,3 is the free Hamiltonian for three identical particles of mass m in contact interaction.
The interaction of particles one and two is independent from that of particles 2 and 3; one can therefore take the limits in every order, but simultaneous contact of three wave functions is too singular and does not provide a self-adjoint operator.
Open chains of N particle in contact interaction at the junctions are stable system described by self-adjoint operators.
Given a chain of N wave function in contact interaction at the junctions, one can add at either end a new particle by a attractive contact In the same way given a chain of N particles in contact interaction at the junctions, one can by a repulsive contact interaction separate the particle at either end and obtain a stable chain of N − 1 wave functions in contact described by a self-adjoint operator.
By a repulsive contact, one can separate a chain on N wave in contact interaction into two chains of N 1 and N 2 particle in contact interaction with N 1 + N 2 = N .
One can describe in this way creation and annihilation of particles. This setting is somehow artificial and in any case very different from the case of dimension 3 where the stable clusters are of three particles (Efimov effect) or of four (the Bose-Einstein condensate) and the cluster of three cannot be obtained from the cluster of four simply by taking away one particle. A rearrangement of the interactions was necessary.

The case of fermions
We have considered so far only the case of bosons which satisfy the Scrödinger equation.
The case of spin 1 2 fermions that satisfy the Pauli equation can be treated similarly. In two dimensions the spin group is abelian, and the square of the Pauli operator is the Schrödinger operator on each component.
By the Pauli principle, one can have contact only between opposite spins. Therefore, the condensate is made of pairs of identical fermions with opposite spin in contact, i.e., of bosons.
Funding Open access funding provided by Scuola Internazionale Superiore di Studi Avanzati -SISSA within the CRUI-CARE Agreement.
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