@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{22076,
  abstract     = {In dimension n > 3 we show the existence of a compactly supported potential in the differentiability class Cα, α < n-3/2, for which the solutions to the linear Schrödinger equation in ℝn, ∂tu = - Δ u + Vu, u(0)=f, fail to satisfy an evolution estimate of the form ||u(t, ̇)||∞ ≤ C |t| -n/2 ||u(0,\̇)||1̇. This contrasts with known results in dimensions n ≤ 3, where a pointwise decay condition on V is generally sufficient to imply dispersive bounds. The obstructions in our example are generated by an expression with scaling law |t|-n+3/2+α, which becomes dominant in the time interval |t| ≪ 1.},
  author       = {Goldberg, Michael and Visan, Monica},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  pages        = {211--238},
  publisher    = {Springer Nature},
  title        = {{A counterexample to dispersive estimates for Schrödinger operators in higher dimensions}},
  doi          = {10.1007/s00220-006-0013-5},
  volume       = {266},
  year         = {2006},
}

@article{2356,
  abstract     = {Recent experimental and theoretical work has shown that there are conditions in which a trapped, low-density Bose gas behaves like the one-dimensional delta-function Bose gas solved years ago by Lieb and Liniger. This is an intrinsically quantum-mechanical phenomenon because it is not necessary to have a trap width that is the size of an atom - as might have been supposed - but it suffices merely to have a trap width such that the energy gap for motion in the transverse direction is large compared to the energy associated with the motion along the trap. Up to now the theoretical arguments have been based on variational - perturbative ideas or numerical investigations. In contrast, this paper gives a rigorous proof of the one-dimensional behavior as far as the ground state energy and particle density are concerned. There are four parameters involved: the particle number, N, transverse and longitudinal dimensions of the trap, r and L, and the scattering length a of the interaction potential. Our main result is that if r/L → 0 and N → ∞ the ground state energy and density can be obtained by minimizing a one-dimensional density functional involving the Lieb-Liniger energy density with coupling constant ∼ a/r 2. This density functional simplifies in various limiting cases and we identify five asymptotic parameter regions altogether. Three of these, corresponding to the weak coupling regime, can also be obtained as limits of a three-dimensional Gross-Pitaevskii theory. We also show that Bose-Einstein condensation in the ground state persists in a part of this regime. In the strong coupling regime the longitudinal motion of the particles is strongly correlated. The Gross-Pitaevskii description is not valid in this regime and new mathematical methods come into play.},
  author       = {Lieb, Élliott and Seiringer, Robert and Yngvason, Jakob},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  number       = {2},
  pages        = {347 -- 393},
  publisher    = {Springer Nature},
  title        = {{One-dimensional behavior of dilute, trapped Bose gases}},
  doi          = {10.1007/s00220-003-0993-3},
  volume       = {244},
  year         = {2004},
}

@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},
}