@article{554,
  abstract     = {We analyse the canonical Bogoliubov free energy functional in three dimensions at low temperatures in the dilute limit. We prove existence of a first-order phase transition and, in the limit (Formula presented.), we determine the critical temperature to be (Formula presented.) to leading order. Here, (Formula presented.) is the critical temperature of the free Bose gas, ρ is the density of the gas and a is the scattering length of the pair-interaction potential V. We also prove asymptotic expansions for the free energy. In particular, we recover the Lee–Huang–Yang formula in the limit (Formula presented.).},
  author       = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  number       = {1},
  pages        = {347--403},
  publisher    = {Springer},
  title        = {{The Bogoliubov free energy functional II: The dilute Limit}},
  doi          = {10.1007/s00220-017-3064-x},
  volume       = {360},
  year         = {2018},
}

@article{1207,
  abstract     = {The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.},
  author       = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  number       = {3},
  pages        = {947 -- 990},
  publisher    = {Springer},
  title        = {{Local law of addition of random matrices on optimal scale}},
  doi          = {10.1007/s00220-016-2805-6},
  volume       = {349},
  year         = {2017},
}

@article{741,
  abstract     = {We prove that a system of N fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical m*. The value of m* is independent of N and turns out to be less than 1. This fact has important implications for the stability of the unitary Fermi gas. We also characterize the domain of the Hamiltonian of this model, and establish the validity of the Tan relations for all wave functions in the domain.},
  author       = {Moser, Thomas and Seiringer, Robert},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  number       = {1},
  pages        = {329 -- 355},
  publisher    = {Springer},
  title        = {{Stability of a fermionic N+1 particle system with point interactions}},
  doi          = {10.1007/s00220-017-2980-0},
  volume       = {356},
  year         = {2017},
}

@article{8493,
  abstract     = {In this paper we study a so-called separatrix map introduced by Zaslavskii–Filonenko (Sov Phys JETP 27:851–857, 1968) and studied by Treschev (Physica D 116(1–2):21–43, 1998; J Nonlinear Sci 12(1):27–58, 2002), Piftankin (Nonlinearity (19):2617–2644, 2006) Piftankin and Treshchëv (Uspekhi Mat Nauk 62(2(374)):3–108, 2007). We derive a second order expansion of this map for trigonometric perturbations. In Castejon et al. (Random iteration of maps of a cylinder and diffusive behavior. Preprint available at arXiv:1501.03319, 2015), Guardia and Kaloshin (Stochastic diffusive behavior through big gaps in a priori unstable systems (in preparation), 2015), and Kaloshin et al. (Normally Hyperbolic Invariant Laminations and diffusive behavior for the generalized Arnold example away from resonances. Preprint available at http://www.terpconnect.umd.edu/vkaloshi/, 2015), applying the results of the present paper, we describe a class of nearly integrable deterministic systems with stochastic diffusive behavior.},
  author       = {Guardia, M. and Kaloshin, Vadim and Zhang, J.},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  pages        = {321--361},
  publisher    = {Springer Nature},
  title        = {{A second order expansion of the separatrix map for trigonometric perturbations of a priori unstable systems}},
  doi          = {10.1007/s00220-016-2705-9},
  volume       = {348},
  year         = {2016},
}

@article{1935,
  abstract     = {We consider Ising models in d = 2 and d = 3 dimensions with nearest neighbor ferromagnetic and long-range antiferromagnetic interactions, the latter decaying as (distance)-p, p > 2d, at large distances. If the strength J of the ferromagnetic interaction is larger than a critical value J c, then the ground state is homogeneous. It has been conjectured that when J is smaller than but close to J c, the ground state is periodic and striped, with stripes of constant width h = h(J), and h → ∞ as J → Jc -. (In d = 3 stripes mean slabs, not columns.) Here we rigorously prove that, if we normalize the energy in such a way that the energy of the homogeneous state is zero, then the ratio e 0(J)/e S(J) tends to 1 as J → Jc -, with e S(J) being the energy per site of the optimal periodic striped/slabbed state and e 0(J) the actual ground state energy per site of the system. Our proof comes with explicit bounds on the difference e 0(J)-e S(J) at small but positive J c-J, and also shows that in this parameter range the ground state is striped/slabbed in a certain sense: namely, if one looks at a randomly chosen window, of suitable size ℓ (very large compared to the optimal stripe size h(J)), one finds a striped/slabbed state with high probability.},
  author       = {Giuliani, Alessandro and Lieb, Élliott and Seiringer, Robert},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  pages        = {333 -- 350},
  publisher    = {Springer},
  title        = {{Formation of stripes and slabs near the ferromagnetic transition}},
  doi          = {10.1007/s00220-014-1923-2},
  volume       = {331},
  year         = {2014},
}

@article{8502,
  abstract     = {The famous ergodic hypothesis suggests that for a typical Hamiltonian on a typical energy surface nearly all trajectories are dense. KAM theory disproves it. Ehrenfest (The Conceptual Foundations of the Statistical Approach in Mechanics. Ithaca, NY: Cornell University Press, 1959) and Birkhoff (Collected Math Papers. Vol 2, New York: Dover, pp 462–465, 1968) stated the quasi-ergodic hypothesis claiming that a typical Hamiltonian on a typical energy surface has a dense orbit. This question is wide open. Herman (Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998). Doc Math 1998, Extra Vol II, Berlin: Int Math Union, pp 797–808, 1998) proposed to look for an example of a Hamiltonian near H0(I)=⟨I,I⟩2 with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian H0(I)+εH1(θ,I,ε) which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.},
  author       = {Kaloshin, Vadim and Saprykina, Maria},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  keywords     = {Mathematical Physics, Statistical and Nonlinear Physics},
  number       = {3},
  pages        = {643--697},
  publisher    = {Springer Nature},
  title        = {{An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension}},
  doi          = {10.1007/s00220-012-1532-x},
  volume       = {315},
  year         = {2012},
}

@article{1459,
  abstract     = {In this paper we explicitly calculate the analogue of the 't Hooft SU (2) Yang-Mills instantons on Gibbons-Hawking multi-centered gravitational instantons, which come in two parallel families: the multi-Eguchi-Hanson, or Ak ALE gravitational instantons and the multi-Taub-NUT spaces, or Ak ALF gravitational instantons. We calculate their energy and find the reducible ones. Following Kronheimer we also exploit the U(1) invariance of our solutions and study the corresponding explicit singular SU (2) magnetic monopole solutions of the Bogomolny equations on flat ℝ3.},
  author       = {Etesi, Gábor and Hausel, Tamas},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  pages        = {275 -- 288},
  publisher    = {Springer Nature},
  title        = {{On Yang-Mills instantons over multi-centered gravitational instantons}},
  doi          = {10.1007/s00220-003-0806-8},
  volume       = {235},
  year         = {2003},
}

@article{2739,
  abstract     = {We define the two dimensional Pauli operator and identify its core for magnetic fields that are regular Borel measures. The magnetic field is generated by a scalar potential hence we bypass the usual A L 2loc condition on the vector potential, which does not allow to consider such singular fields. We extend the Aharonov-Casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite total variation. One of the key technical tools is a weighted L 2 estimate on a singular integral operator.},
  author       = {Erdös, László and Vougalter, Vitali},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  number       = {2},
  pages        = {399 -- 421},
  publisher    = {Springer},
  title        = {{Pauli operator and Aharonov–Casher theorem¶ for measure valued magnetic fields}},
  doi          = {10.1007/s002200100585},
  volume       = {225},
  year         = {2002},
}

@article{2347,
  abstract     = {We consider the ground state properties of an inhomogeneous two-dimensional Bose gas with a repulsive, short range pair interaction and an external confining potential. In the limit when the particle number N is large but ρ̅a 2 is small, where ρ̅ is the average particle density and a the scattering length, the ground state energy and density are rigorously shown to be given to leading order by a Gross–Pitaevskii (GP) energy functional with a coupling constant g~1/|1n(ρ̅a 2)|. In contrast to the 3D case the coupling constant depends on N through the mean density. The GP energy per particle depends only on Ng. In 2D this parameter is typically so large that the gradient term in the GP energy functional is negligible and the simpler description by a Thomas–Fermi type functional is adequate.},
  author       = {Lieb, Élliott and Seiringer, Robert and Yngvason, Jakob},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  number       = {1},
  pages        = {17 -- 31},
  publisher    = {Springer},
  title        = {{A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas}},
  doi          = {10.1007/s002200100533},
  volume       = {224},
  year         = {2001},
}

@article{2348,
  abstract     = {This paper concerns the asymptotic ground state properties of heavy atoms in strong, homogeneous magnetic fields. In the limit when the nuclear charge Z tends to ∞ with the magnetic field B satisfying B ≫ Z4/3 all the electrons are confined to the lowest Landau band. We consider here an energy functional, whose variable is a sequence of one-dimensional density matrices corresponding to different angular momentum functions in the lowest Landau band. We study this functional in detail and derive various interesting properties, which are compared with the density matrix (DM) theory introduced by Lieb, Solovej and Yngvason. In contrast to the DM theory the variable perpendicular to the field is replaced by the discrete angular momentum quantum numbers. Hence we call the new functional a discrete density matrix (DDM) functional. We relate this DDM theory to the lowest Landau band quantum mechanics and show that it reproduces correctly the ground state energy apart from errors due to the indirect part of the Coulomb interaction energy.},
  author       = {Hainzl, Christian and Seiringer, Robert},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  number       = {1},
  pages        = {229 -- 248},
  publisher    = {Springer},
  title        = {{A discrete density matrix theory for atoms in strong magnetic fields}},
  doi          = {10.1007/s002200100373},
  volume       = {217},
  year         = {2001},
}

@article{8525,
  abstract     = {Let M be a smooth compact manifold of dimension at least 2 and Diffr(M) be the space of C r smooth diffeomorphisms of M. Associate to each diffeomorphism f;isin; Diffr(M) the sequence P n (f) of the number of isolated periodic points for f of period n. In this paper we exhibit an open set N in the space of diffeomorphisms Diffr(M) such for a Baire generic diffeomorphism f∈N the number of periodic points P n f grows with a period n faster than any following sequence of numbers {a n } n ∈ Z + along a subsequence, i.e. P n (f)>a ni for some n i →∞ with i→∞. In the cases of surface diffeomorphisms, i.e. dim M≡2, an open set N with a supergrowth of the number of periodic points is a Newhouse domain. A proof of the man result is based on the Gontchenko–Shilnikov–Turaev Theorem [GST]. A complete proof of that theorem is also presented.},
  author       = {Kaloshin, Vadim},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  keywords     = {Mathematical Physics, Statistical and Nonlinear Physics},
  pages        = {253--271},
  publisher    = {Springer Nature},
  title        = {{Generic diffeomorphisms with superexponential growth of number of periodic orbits}},
  doi          = {10.1007/s002200050811},
  volume       = {211},
  year         = {2000},
}

@article{2729,
  abstract     = {We give the leading order semiclassical asymptotics for the sum of the negative eigenvalues of the Pauli operator (in dimension two and three) with a strong non-homogeneous magnetic field. As in [LSY-II] for homogeneous field, this result can be used to prove that the magnetic Thomas-Fermi theory gives the leading order ground state energy of large atoms. We develop a new localization scheme well suited to the anisotropic character of the strong magnetic field. We also use the basic Lieb-Thirring estimate obtained in our companion paper [ES-I].},
  author       = {Erdös, László and Solovej, Jan},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  number       = {3},
  pages        = {599 -- 656},
  publisher    = {Springer},
  title        = {{Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields, II. Leading order asymptotic estimates}},
  doi          = {10.1007/s002200050181},
  volume       = {188},
  year         = {1997},
}

@article{2724,
  abstract     = {We study the generalizations of the well-known Lieb-Thirring inequality for the magnetic Schrödinger operator with nonconstant magnetic field. Our main result is the naturally expected magnetic Lieb-Thirring estimate on the moments of the negative eigenvalues for a certain class of magnetic fields (including even some unbounded ones). We develop a localization technique in path space of the stochastic Feynman-Kac representation of the heat kernel which effectively estimates the oscillatory effect due to the magnetic phase factor.},
  author       = {Erdös, László},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  number       = {3},
  pages        = {629 -- 668},
  publisher    = {Springer},
  title        = {{Magnetic Lieb-Thirring inequalities}},
  doi          = {10.1007/BF02099152},
  volume       = {170},
  year         = {1995},
}

@article{2722,
  abstract     = {A version of the one-dimensional Rayleigh gas is considered: a point particle of mass M (molecule), confined to the unit interval [0,1], is surrounded by an infinite ideal gas of point particles of mass 1 (atoms). The molecule interacts with the atoms and with the walls via elastic collision. Central limit theorems are proved for a wide class of additive functionals of this system (e.g. the number of collisions with the walls and the total length of the molecular path).},
  author       = {Erdös, László and Tuyen, Dao},
  issn         = {0010-3616},
  journal      = {Communications in Mathematical Physics},
  number       = {3},
  pages        = {451 -- 466},
  publisher    = {Springer},
  title        = {{Central limit theorems for the one-dimensional Rayleigh gas with semipermeable barriers}},
  doi          = {10.1007/BF02099260},
  volume       = {143},
  year         = {1992},
}