Arbitrary harmonic functions as Bose--Einstein condensates

Technical Summary: Arbitrary Harmonic Functions as Bose–Einstein Condensates

Problem Statement
The paper investigates the nature of Bose–Einstein condensates (BECs) that can arise in the thermodynamic limit of an ideal Bose gas when the boundary conditions for the Laplacian on the domain boundary are freely chosen. While previous observations (e.g., in [5]) indicated that infinite-volume ideal Bose gases admit many different KMS states characterized by arbitrary positive sesquilinear forms in the two-point function—corresponding to harmonic functions describing condensates—it remained an open question whether such states could be rigorously obtained via the standard thermodynamic limit procedure (taking the limit of Gibbs states of finite-volume systems). Furthermore, while non-translation-invariant KMS states with linear condensates were known to exist [6], the possibility of realizing condensates described by arbitrary harmonic functions, or multiple condensates simultaneously, through specific boundary conditions had not been established.

Methodology
The authors address this problem by constructing a specific sequence of positive self-adjoint realizations of the Laplacian, denoted $-\Delta_{\Omega_j}^\Phi$, on a sequence of bounded domains $\Omega_j \subset \mathbb{R}^d$ that exhaust $\mathbb{R}^d$.

1. Construction of the Operator:
   The core of the methodology involves defining the inverse of the Laplacian on a domain $\Omega$ via a perturbation of the Dirichlet Laplacian inverse. Given a sequence of harmonic functions $\Phi = \{\phi_k\}_k$ on $\mathbb{R}^d$ satisfying $\sum_k |\phi_k(x)|^2 < \infty$, the operator is defined as:
   $$(-\Delta_{\Omega}^\Phi)^{-1} := \left( (-\Delta_{\Omega}^0)^{-1} + \sum_k |\chi_\Omega \phi_k\rangle \langle \phi_k \chi_\Omega| \right)^{-1}$$
   where $-\Delta_{\Omega}^0$ is the Dirichlet Laplacian and $\chi_\Omega$ is the characteristic function of $\Omega$. This construction ensures that the new operator is a positive self-adjoint extension of the Laplacian on $C_0^\infty(\Omega)$.

2. Boundary Conditions:
   The authors derive the explicit boundary conditions associated with this operator. They show that functions in the domain of $-\Delta_{\Omega}^\Phi$ satisfy a generalized Robin-type boundary condition involving the Dirichlet-to-Neumann operator $H_\Omega$ and a specific operator $r_\Omega^\Phi$ determined by the harmonic functions $\phi_k$. Specifically, the normal derivative $\partial_n f$ on the boundary is related to the boundary values of $f$ and the projection of $f$ onto the span of the harmonic functions.

3. Convergence Analysis:
   To establish the thermodynamic limit, the authors prove a strong resolvent convergence result (Theorem 2). They demonstrate that for any sequence of non-negative self-adjoint extensions of the Laplacian on domains $\Omega_j$ converging to $\mathbb{R}^d$, the operators $F(-\Delta_{\Omega_j})$ converge strongly to $F(-\Delta)$ on $L^2(\mathbb{R}^d)$ for any bounded continuous function $F$. This general result is then applied to the specific function $F(x) = (e^{\beta x} - 1)^{-1} - (\beta x)^{-1}$, which relates to the Bose gas two-point function.

Key Results
The main result is presented in Theorem 1. The authors prove that for any sequence of harmonic functions $\Phi = \{\phi_k\}_k$ (satisfying the square-summability condition), there exists a sequence of boundary conditions such that the two-point function of the ideal Bose gas in the thermodynamic limit converges to:
$$\lim_{j\to\infty} \left\langle f \left| \frac{1}{e^{-\beta \Delta_{\Omega_j}^\Phi} - 1} \right| g \right\rangle = \int_{\mathbb{R}^d} \frac{\hat{f}(p)\overline{\hat{g}(p)}}{e^{\beta|p|^2} - 1} dp + \frac{1}{\beta} \sum_k \left( \int_{\mathbb{R}^d} f \phi_k \right) \left( \int_{\mathbb{R}^d} \overline{g \phi_k} \right)$$
Key features of this result include:

• Arbitrary Condensates: The system supports an arbitrary number (finite or infinite) of Bose–Einstein condensates.
• Harmonic Profiles: Each condensate is described by an arbitrary harmonic function $\phi_k$. The density of the $k$-th condensate is proportional to $|\phi_k(x)|^2$.
• Zero Chemical Potential: The result holds at zero chemical potential. The authors note that the result extends to small negative chemical potentials that vanish sufficiently fast as the volume increases.
• Dimensional Constraints: In dimensions $d \in \{1, 2\}$, the convergence requires the additional assumption that the Fourier transforms of the test functions vanish at zero ($\hat{f}(0) = \hat{g}(0) = 0$) to ensure integrability of the free gas term at low momenta.

Significance and Claims
The paper claims to resolve the question of whether the diverse KMS states observed in infinite volume can be realized as limits of finite-volume Gibbs states. By explicitly constructing the boundary conditions, the authors show that:

1. It is possible to engineer a system where the condensate wavefunctions are not restricted to constants or linear functions but can be any harmonic functions.
2. Multiple condensates can coexist in the thermodynamic limit, a phenomenon distinct from the "multiple condensates" observed in finite volumes with Dirichlet boundary conditions (where the effect vanishes in the limit of the two-point function).
3. The eigenvalues of the constructed Laplacians must decrease as $(\int_\Omega |\phi_k|^2)^{-1}$ as the domain expands, ensuring that the spectral projections associated with these small eigenvalues yield finite, non-zero contributions to the condensate density in the limit.

The authors maintain a modest scope, focusing strictly on the mathematical existence of such states within the framework of ideal Bose gases and the thermodynamic limit, without proposing experimental realizations or extending the results to interacting systems. The work relies on the spectral theory of self-adjoint extensions and the convergence of resolvents to bridge the gap between finite-volume boundary value problems and infinite-volume statistical mechanics.