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Robert Seiringer

Publications and source records attributed to Robert Seiringer.

At least 19 recordsLinked to original sources
arXiv2026

The Huang--Yang formula for a two-dimensional Fermi gas: upper bound

We compute an upper bound on the ground state energy of a dilute two-dimensional Fermi gas with repulsive short-range interactions. Our bound can be viewed as the two-dimensional analogue of a formula derived by Huang and Yang in the three-dimensional case. It captures the first three terms in an asymptotic expansion for small $\varrho a^2$, where $\varrho$ denotes the density and $a$ the scattering length of the interaction potential.

math-ph↗
arXiv2026

Liquid Drop Model for Nuclear Matter in the Low Density Limit

We consider the liquid drop model with a positive background density in the thermodynamic limit. We prove a two-term asymptotics for the ground state energy per unit volume in the dilute limit. Our proof justifies the expectation that optimal configurations consist of droplets of unit size that arrange themselves according to minimizers for the Jellium problem for point particles. In particular, we provide the first rigorous derivation of what is known as the gnocchi phase in astrophysics.

math-ph↗
arXiv2026

Arbitrary harmonic functions as Bose--Einstein condensates

We show that a suitable choice of boundary conditions for the Laplacian allows for the appearance of an an arbitrary number of condensates, described by arbitrary harmonic functions, in the thermodynamic limit of an ideal Bose gas.

math-ph↗
arXiv2025

Dyson expansion for form-bounded perturbations and applications to the polaron problem

We present an abstract Dyson expansion for perturbations that are merely relatively form-bounded, and apply it to the polaron problem. For a large class of polaron-type models, including the Fr\"ohlich and Nelson models, we prove that the vacuum expectation value of the heat semi-group is a completely monotone function of the square of the total momentum. Consequently, the ground state energy is a concave function of the square of the momentum, a result recently proved for the Fr\"ohlich model in \cite{polzer} using a probabilistic approach via Wiener integrals.

math-ph↗
arXiv2025

Bound excited states of Fröhlich polarons in one dimension

The one-dimensional Fröhlich model describing the motion of a single electron interacting with optical phonons is a paradigmatic model of quantum many-body physics. We predict the existence of an arbitrarily large number of bound excited states in the strong coupling limit and calculate their excitation energies. Numerical simulations of a discretized model demonstrate the complete amelioration of the projector Monte Carlo sign problem by walker annihilation in an infinite Hilbert space. They reveal the threshold for the occurrence of the first bound excited states at a value of $α\approx 1.73$ for the dimensionless coupling constant. This puts the threshold into the regime of intermediate interaction strength. We find a significant spectral weight and increased phonon number of the bound excited state at threshold.

math-ph↗
arXiv2025

The Huang-Yang conjecture for the low-density Fermi gas

Our work establishes a three-term asymptotic expansion of the ground state energy of a dilute gas of spin $1/2$ fermions with repulsive short-range interactions, validating a formula predicted by Huang and Yang in 1957. The formula is universal in the sense that it holds for a large class of interaction potentials and depends on those only via their scattering length. We have recently proved an upper bound on the ground state energy of the desired form, and the present work completes the program by proving the matching lower bound.

math-ph↗
arXiv2025

Enhanced Superconductivity at a Corner for the Linear BCS Equation

We consider the critical temperature for superconductivity, defined via the linear BCS equation. We prove that at weak coupling the critical temperature for a sample confined to a quadrant in two dimensions is strictly larger than the one for a half-space, which in turn is strictly larger than the one for $\mathbb{R}^2$. Furthermore, we prove that the relative difference of the critical temperatures vanishes in the weak coupling limit.

math-ph↗
arXiv2025

BCS Critical Temperature on Half-Spaces

We study the BCS critical temperature on half-spaces in dimensions $d=1,2,3$ with Dirichlet or Neumann boundary conditions. We prove that the critical temperature on a half-space is strictly higher than on $\mathbb{R}^d$, at least at weak coupling in $d=1,2$ and weak coupling and small chemical potential in $d=3$. Furthermore, we show that the relative shift in critical temperature vanishes in the weak coupling limit.

math-ph↗
arXiv2024

Stability estimate for the Lane-Emden inequality

The Lane-Emden inequality controls $\iint_{\mathbb{R}^{2d}}ρ(x)ρ(y)|x-y|^{-λ}\,dx\,dy$ in terms of the $L^1$ and $L^p$ norms of $ρ$. We provide a remainder estimate for this inequality in terms of a suitable distance of $ρ$ to the manifold of optimizers.

math.AP↗
arXiv2024

The Huang-Yang formula for the low-density Fermi gas: upper bound

We study the ground state energy of a gas of spin $1/2$ fermions with repulsive short-range interactions. We derive an upper bound that agrees, at low density $\rho$, with the Huang-Yang conjecture. The latter captures the first three terms in an asymptotic low-density expansion, and in particular the Huang-Yang correction term of order $\rho^{7/3}$. Our trial state is constructed using an adaptation of the bosonic Bogoliubov theory to the Fermi system, where the correlation structure of fermionic particles is incorporated by quasi-bosonic Bogoliubov transformations. In the latter, it is important to consider a modified zero-energy scattering equation that takes into account the presence of the Fermi sea, in the spirit of the Bethe-Goldstone equation.

math-ph↗
arXiv2024

Pressure of a dilute spin-polarized Fermi gas: Upper bound

We prove an upper bound on the pressure of a dilute fully spin-polarized Fermi gas capturing the leading correction to the pressure of a free gas resulting from repulsive interactions. This correction is of order $a^3ρ^{8/3}$, with $a$ the $p$-wave scattering length of the interaction and $ρ$ the particle density, depends on the temperature and matches the corresponding lower bound of [arXiv:2307.01113].

math-ph↗
arXiv2024

Ground state energy of the dilute spin-polarized Fermi gas: Lower bound

We prove a lower bound on the ground state energy of the dilute spin-polarized Fermi gas capturing the leading correction to the kinetic energy resulting from repulsive interactions. This correction depends on the $p$-wave scattering length of the interaction and matches the corresponding upper bound in [J. Funct. Anal. 286.7 (2024), p. 110320].

math-ph↗
arXiv2024

The free energy of dilute Bose gases at low temperatures

We consider a low density Bose gas interacting through a repulsive potential in the thermodynamic limit. We justify, as a rigorous lower bound, a Lee--Huang--Yang type formula for the free energy at suitably low temperatures, where the modified excitation spectrum leads to a second order correction of the same order as the Lee--Huang--Yang correction to the ground state energy.

math-ph↗
arXiv2023

A simple approach to Lieb--Thirring type inequalities

In \cite{Nam} Nam proved a Lieb--Thirring Inequality for the kinetic energy of a fermionic quantum system, with almost optimal (semi-classical) constant and a gradient correction term. We present a stronger version of this inequality, with a much simplified proof. As a corollary we obtain a simple proof of the original Lieb--Thirring inequality.

math-ph↗
arXiv2023

Pressure of a dilute spin-polarized Fermi gas: Lower bound

We consider a dilute spin-polarized Fermi gas at positive temperature in dimensions $d\in\{1,2,3\}$. We show that the pressure of the interacting gas is bounded from below by that of the free gas plus, to leading order, an explicit term of order $a^dρ^{2+2/d}$, where $a$ is the $p$-wave scattering length of the repulsive interaction and $ρ$ is the particle density. The results are valid for a wide range of repulsive interactions, including that of a hard core, and uniform in temperatures at most of the order of the Fermi temperature. A central ingredient in the proof is a rigorous implementation of the fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237--260).

math-ph↗
arXiv2023

Boundary Superconductivity in the BCS Model

We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg-Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.

math-ph↗
arXiv2023

Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential

Recently the leading order of the correlation energy of a Fermi gas in a coupled mean-field and semiclassical scaling regime has been derived, under the assumption of an interaction potential with a small norm and with compact support in Fourier space. We generalize this result to large interaction potentials, requiring only $|\cdot| \hat{V} \in \ell^1 (\mathbb{Z}^3)$. Our proof is based on approximate, collective bosonization in three dimensions. Significant improvements compared to recent work include stronger bounds on non-bosonizable terms and more efficient control on the bosonization of the kinetic energy.

math-ph↗