Caltech Home > PMA Home > People > Rupert L. Frank
open search form
Caltech
The Division of Physics,
Mathematics and Astronomy
open menu close menu

Rupert L. Frank

Visiting Associate in Mathematics
Contact information for Rupert L. Frank
Contact Method Value
Mail Code: MC 253-37
Office: 178 Sloan
Phone: 626-395-4323
Research Website
Dipl., Ludwig-Maximilians-Universitat Munich, 2003; Ph.D., Royal Institute of Technology, 2007. Caltech, 2013-2021.
Research Areas: Mathematics

Rupert works in Analysis, PDE and Mathematical Physics. His research aims at developing new mathematical tools that enhance our qualitative and quantitative understanding of complex phenomena occurring in nature. Of particular interest are problems related to Quantum Physics.


Some recorded talks:

Lieb-Thirring bounds and other inequalities for orthonormal functions (Probability and Analysis Webinair, February 22, 2021)

Which magnetic fields support a zero mode? (Geometric Analysis Seminar, January 12, 2021)

Inequalities for L^p norms that sharpen the triangle inequality (Corona Seminar, December 10, 2020)

Recent results and open problems on Lieb-Thirring inequalities (Math Phys Seminar, Rutgers, November 18, 2020)

Minimal magnetic fields supporting a zero mode (Ari Laptev's 70th birthday conference, August 12, 2020)

Sharp Weyl Laws in 3d with rough potentials (Spectral geometry in the clouds Seminar, July 6, 2020)

A `liquid-solid' phase transition in a simple model for swarming (One World PDE Seminar, June 2, 2020)

Minicourse: A microscopic derivation of Ginzburg-Landau theory Part I, Part II, Part III (IHES, July 2016)

Stability estimates for the lowest eigenvalue of a Schrödinger operator (Banff, August 1, 2013)

Uniqueness and Nondegeneracy of Ground States for Non-Local Equations (Institute of Advanced Study, Princeton, October 19, 2012)


Editorial activity

Communications in Contemporary Mathematics

Journal in Mathematical Physics

Journal of Spectral Theory

SIAM Journal on Mathematical Analysis

Selected Publications

  1. On the scattering theory of the Laplacian with a periodic boundary condition. I. Existence of wave operators. Doc. Math. 8 (2003), 547 - 565. Article
  2. (with R. G. Shterenberg) On the scattering theory of the Laplacian with a periodic boundary condition. II. Additional channels of scattering. Doc. Math. 9 (2004), 57 - 77. Article
  3. On the Laplacian in the halfspace with a periodic boundary condition. Ark. Mat. 44 (2006), no. 2, 277 - 298. Article - Preprint
  4. On the tunneling effect for magnetic Schrödinger operators in antidot lattices. Asymptot. Anal. 48 (2006), no. 1-2, 91 - 120. Article - Preprint
  5. (with R. G. Shterenberg) On the spectrum of partially periodic operators. In: Operator Theory, Analysis and Mathematical Physics, J. Janas, et al. (eds.), 35 - 50, Oper. Theory Adv. Appl. 174, Birkhäuser, Basel, 2007. Article - Preprint
  6. (with O. Safronov) Absolutely continuous spectrum of a class of random nonergodic Schrödinger operators. Int. Math. Res. Not. (2005), no. 42, 2559 - 2577. Article - Preprint
  7. (with P. Exner) Absolute continuity of the spectrum for periodically modulated leaky wires in R^3. Ann. Henri Poincaré 8 (2007), no. 2, 241 - 263. Article - Preprint
  8. (with T. Ekholm) On Lieb-Thirring inequalities for Schrödinger operators with virtual level. Comm. Math. Phys. 264 (2006), no. 3, 725 - 740. Article - Preprint
  9. On the asymptotic number of edge states for magnetic Schrödinger operators. Proc. London Math. Soc. (3) 95 (2007), no. 1, 1 - 19. Article - Preprint
  10. (with A. Laptev, E. H. Lieb and R. Seiringer) Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys. 77 (2006), 309 - 316. Article - Preprint
  11. (with E. H. Lieb and R. Seiringer) Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. J. Amer. Math. Soc. 21 (2008), no. 4, 925 - 950. Article - Preprint
  12. (with E. H. Lieb and R. Seiringer) Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value. Comm. Math. Phys. 275 (2007), no. 2, 479 - 489. Article - Preprint
  13. (with T. Ekholm) Lieb-Thirring inequalities on the half-line with critical exponent. J. Eur. Math. Soc. 10 (2008), no. 3, 739 - 755. Article - Preprint
  14. (with A. M. Hansson) Eigenvalue estimates for the Aharonov-Bohm operator in a domain. In: Methods of Spectral Analysis in Mathematical Physics, J. Janas, et al. (eds.), 35 - 50, Oper. Theory Adv. Appl. 186, Birkhäuser, Basel, 2008. Article - Preprint
  15. (with H. Siedentop and S. Warzel) The ground state energy of heavy atoms: relativistic lowering of the leading energy correction. Comm. Math. Phys. 278 (2008), no. 2, 549 - 566. Article - Preprint
  16. (with C. Hainzl, S. Naboko and R. Seiringer) The critical temperature in the BCS model at weak coupling. J. Geom. Anal. 17 (2007), no. 4, 559 - 568. Article - Preprint
  17. (with R. D. Benguria and M. Loss) The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space. Math. Res. Lett. 15 (2008), no. 4, 613 - 622. Article - Preprint
  18. (with E. H. Lieb, R. Seiringer and H. Siedentop) Müller's exchange-correlation energy in density-matrix-functional theory. Phys. Rev. A 76 (2007), 052517. Article - Preprint
  19. (with A. Laptev and S. Molchanov) Eigenvalue estimates for magnetic Schrödinger operators in domains. Proc. Amer. Math. Soc. 136 (2008), no. 12, 4245 - 4255. Article - Preprint
  20. (with B. Simon and T. Weidl) Eigenvalue bounds for perturbations of Schrödinger operators and Jacobi matrices with regular ground states. Comm. Math. Phys. 282 (2008), no. 1, 199 - 208. Article - Preprint
  21. (with M. Loss and T. Weidl) Polya's conjecture in the presence of a constant magnetic field. J. Eur. Math. Soc. 11 (2009), 1365 - 1383. Article - Preprint
  22. (with E. H. Lieb and R. Seiringer) Number of bound states of Schrödinger operators with matrix-valued potentials. Lett. Math. Phys. 82 (2007), 107 - 116. Article - Preprint
  23. (with T. Ekholm and H. Kovarik) Eigenvalue estimates for Schrödinger operators on metric trees. Adv. Math. 226 (2011), 5165 - 5197. Article - Preprint
  24. (with T. Ekholm and H. Kovarik) Remarks about Hardy inequalities on metric trees. In: Analysis on graphs and its applications, P. Exner, et al. (eds.), 369 - 379, Proc. Sympos. Pure Math. 77, Amer. Math. Soc., Providence, RI, 2008. Preprint
  25. (with A. Laptev) Spectral inequalities for Schrödinger operators with surface potentials. In: Spectral theory of differential operators, T. Suslina and D. Yafaev (eds.), 91 - 102, Amer. Math. Soc. Transl. Ser. 2, 225 (2008). Preprint
  26. (with R. Seiringer) Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255 (2008), 3407 - 3430. Article - Preprint
  27. (with H. Siedentop and S. Warzel) The energy of heavy atoms according to Brown and Ravenhall: the Scott correction. Doc. Math. 14 (2009), 463 - 516. Article - Preprint
  28. (with J. Breuer) Singular spectrum for radial trees. Rev. Math. Phys. 21 (2009), no. 7, 1 - 17. Article - Preprint
  29. A simple proof of Hardy-Lieb-Thirring inequalities. Comm. Math. Phys. 290 (2009), no. 2, 789 - 800. Article - Preprint
  30. A note on low energy scattering for homogeneous long-range potentials. Ann. Henri Poincaré 10 (2009), no. 3, 573 - 575. Article - Preprint
  31. Remarks on eigenvalue estimates and semigroup domination. In: Spectral and Scattering Theory for Quantum Magnetic Systems, P. Briet, et al. (eds.), 63 - 86, Contemp. Math. 500, Amer. Math. Soc., Providence, RI, 2009. Preprint
  32. (with A. Laptev and R. Seiringer) A sharp bound on eigenvalues of Schrödinger operators on the halfline with complex-valued potentials. In: Spectral Theory and Analysis, J. Janas et al. (eds.), 39 - 44, Oper. Theory Adv. Appl. 214, Birkhäuser, Basel, 2011. Article - Preprint
  33. (with S. Morozov and S. Vugalter) Weakly coupled bound states of Pauli operators. Calc. Var. Partial Differential Equations 40 (2010), no. 1 - 2, 253 - 271. Article - Preprint
  34. (with E. H. Lieb) Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality. Calc. Var. Partial Differential Equations 39 (2010), no. 1 - 2, 85 - 99. Article - Preprint
  35. (with R. Seiringer) Sharp fractional Hardy inequalities in half-spaces. In: Around the research of Vladimir Maz'ya, A. Laptev (ed.), 161 - 167, International Mathematical Series 11 (2010). Article - Preprint
  36. (with A. Laptev) Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group. Int. Math. Res. Not. 15 (2010), 2889 - 2902. Article - Preprint
  37. (with E. H. Lieb and R. Seiringer) Equivalence of Sobolev inequalities and Lieb-Thirring inequalities. In: XVIth International Congress on Mathematical Physics, Proceedings of the ICMP held in Prague, August 3-8, 2009, P. Exner (ed.), 523-535, World Scientific, Singapore, 2010. Article - Preprint
  38. (with B. Simon) Critical Lieb-Thirring bounds in gaps and the generalized Nevai conjecture for finite gap Jacobi matrices. Duke Math. J. 157 (2011), no. 3, 461 - 493. Article - Preprint
  39. (with E. H. Lieb) Spherical reflection positivity and the Hardy-Littlewood-Sobolev inequality. In: Concentration, Functional Inequalities and Isoperimetry, C. Houdré et al. (eds.), Contemp. Math. 545, Amer. Math. Soc., Providence, RI, 2011. Article - Preprint
  40. (with E. H. Lieb, R. Seiringer and L. E. Thomas) Bi-polaron and N-polaron binding energies. Phys. Rev. Lett. 104 (2010), 210402. Article - Preprint
  41. (with E. H. Lieb, R. Seiringer and L. E. Thomas) Stability and absence of binding for multi-polaron systems. Publ. Math. IHES 113 (2011), no. 1, 39 - 67. Article - Preprint
  42. Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc. 43 (2011), no. 4, 745 - 750. Article - Preprint
  43. (with E. H. Lieb) Sharp constants in several inequalities on the Heisenberg group. Ann. of Math. 176 (2012), no. 1, 349 - 381. Article - Preprint
  44. (with E. Lenzmann) Uniqueness and nondegeneracy of ground states for (-Δ)s Q + Q - Qα+1 = 0 in R. Acta Math. 210 (2013), no. 2, 261 - 318. Article - Preprint
  45. (with E. H. Lieb, R. Seiringer and L. E. Thomas) Binding, stability and non-binding of multi-polaron systems. In: Mathematical results in quantum physics, Proceedings of the QMath11 Conference, P. Exner (ed.), 21 - 32, World Scientific, Singapore, 2011. Article - Preprint
  46. (with L. Geisinger) Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain. In: Mathematical results in quantum physics, Proceedings of the QMath11 Conference, P. Exner (ed.), 138 - 147, World Scientific, Singapore, 2011. Article - Preprint
  47. (with E. H. Lieb) A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality. In: Spectral Theory, Function Spaces and Inequalities, B. M. Brown et al. (eds.), 55 - 67, Oper. Theory Adv. Appl. 219, Birkhäuser, Basel, 2012. Article - Preprint
  48. (with D. Lenz and D. Wingert) Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory. J. Funct. Anal. 266 (2014), no. 8, 4765 - 4808. Article - Preprint
  49. (with R. Olofsson) Eigenvalue bounds for Schrödinger operators with a homogeneous magnetic field. Lett. Math. Phys. 97 (2011), no. 3, 227 - 241, Article - Preprint
  50. (with M. Lewin, E. H. Lieb and R. Seiringer) Energy cost to make a hole in the Fermi sea. Phys. Rev. Lett. 106 (2011), 150402. Article - Preprint
  51. (with C. Hainzl, R. Seiringer and J.-P. Solovej) Microscopic derivation of Ginzburg-Landau theory. J. Amer. Math. Soc. 25 (2012), no. 3, 667 - 713. Article - Preprint
  52. (with M. Loss) Hardy-Sobolev-Maz'ya inequalities for arbitrary domains. J. Math. Pures Appl. 97 (2012), 39 - 54. Article - Preprint
  53. (with C. Hainzl, R. Seiringer and J.-P. Solovej) Derivation of Ginzburg-Landau theory for a one-dimensional system with contact interaction. In: Operator Methods in Mathematical Physics, J. Janas et al. (eds.), 57 - 88, Oper. Theory Adv. Appl. 227, Birkhäuser, Basel, 2013. Article - Preprint
  54. (with E. H. Lieb) Possible lattice distortions in the Hubbard model for graphene. Phys. Rev. Lett. 107 (2011), 066801. Article - Preprint
  55. (with L. Geisinger) Refined semiclassical asymptotics for fractional powers of the Laplace operator. J. Reine Angew. Math. 712 (2016), 1 - 37. Article - Preprint
  56. (with E. H. Lieb and R. Seiringer) Binding of polarons and atoms at threshold. Comm. Math. Phys. 313 (2012), no. 2, 405 - 424. Article - Preprint
  57. (with H. D. Cornean, S. Fournais and B. Helffer) Sharp trace asymptotics for a class of 2D-magnetic operators. Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2457 - 2513. Article - Preprint
  58. (with M. Lewin, E. H. Lieb and R. Seiringer) A positive density analogue of the Lieb-Thirring inequality. Duke Math. J. 162 (2013), no. 3, 435 - 495. Article - Preprint
  59. (with H. Kovarik) Heat kernels of metric trees and applications. SIAM J. Math. Anal. 45 (2013), no. 3, 1027 - 1046. Article - Preprint
  60. (with E. H. Lieb) Entropy and the uncertainty principle. Ann. Henri Poincaré 13 (2012), no. 8, 1711 - 1717. Article - Preprint
  61. On the uniqueness of ground states of non-local equations. Journées Équations aux Dérivées Partielles 2011, Exp. No. V, 10 pp. Article - Preprint
  62. (with B. Dyda) Fractional Hardy-Sobolev-Maz'ya inequality for domains. Studia Math. 208 (2012), no. 2, 151 - 166. Article - Preprint
  63. (with R. Seiringer) Lieb-Thirring inequality for a model of particles with point interactions. J. Math. Phys. 53 (2012), no. 9, 095201. Article - Preprint
  64. (with E. H. Lieb and R. Seiringer) Symmetry of bipolaron bound states for small Coulomb repulsion. Comm. Math. Phys. 319 (2013), no. 2, 557 - 573. Article - Preprint
  65. (with E. H. Lieb) Extended quantum conditional entropy and quantum uncertainty inequalities. Comm. Math. Phys. 323 (2013), no. 2, 487 - 495. Article - Preprint
  66. (with S. Chen and T. Weth) Remainder terms in the fractional Sobolev inequality. Indiana Univ. Math. J. 62 (2013), 1381 - 1397. Article - Preprint
  67. Cwikel's theorem and the CLR inequality. J. Spectr. Theory 4 (2014), no. 1, 1 - 21. Article - Preprint
  68. (with L. Geisinger) Semi-classical analysis of the Laplace operator with Robin boundary conditions. Bull. Math. Sci. 2 (2012), no. 2, 281 - 319. Article - Preprint
  69. (with C. Hainzl, R. Seiringer and J.-P. Solovej) Microscopic derivation of the Ginzburg-Landau model. In: XVIIth International Congress on Mathematical Physics, Proceedings of the ICMP held in Aalborg, August 6-11, 2012, A. Jensen (ed.), 575-583, World Scientific, Singapore, 2013. Article - Preprint
  70. (with E. H. Lieb, R. Seiringer and L. E. Thomas) Ground state properties of multi-polaron systems. In: XVIIth International Congress on Mathematical Physics, Proceedings of the ICMP held in Aalborg, August 6-11, 2012, A. Jensen (ed.), 477-485, World Scientific, Singapore, 2013. Article - Preprint
  71. (with E. A. Carlen and E. H. Lieb) Stability estimates for the lowest eigenvalue of a Schrödinger operator. Geom. Funct. Anal. 24 (2014), no. 1, 63 - 84. Article - Preprint
  72. (with J. Bellazzini, E.H. Lieb and R. Seiringer) Existence of ground states for negative ions at the binding threshold. Rev. Math. Phys. 26 (2014), no. 1, 1350021. Article - Preprint
  73. (with E. Lenzmann and L. Silvestre) Uniqueness of radial solutions for the fractional Laplacian. Comm. Pure Appl. Math. 69 (2016), no. 9, 1671 - 1726. Article - Preprint
  74. (with L. Geisinger) The ground state energy of a polaron in a strong magnetic field. Comm. Math. Phys. 338 (2015), no. 1, 1 - 29. Article - Preprint
  75. (with M. Lewin, E. H. Lieb and R. Seiringer) Strichartz inequality for orthonormal functions. J. Eur. Math. Soc. 16 (2014), no. 7, 1507 - 1526. Article - Preprint
  76. (with E. H. Lieb) Monotonicity of a relative Rényi entropy. J. Math. Phys. 54 (2013), 122201. Article - Preprint
  77. (with J. Bellazzini and N. Visciglia) Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems. Math. Ann. 360 (2014), no. 3-4, 653 - 673. Article - Preprint
  78. (with B. Schlein) Dynamics of a strongly coupled polaron. Lett. Math. Phys. 104 (2014), no. 8, 911 - 929. Article - Preprint
  79. (with M. d. M. Gonzalez, D. D. Monticelli and J. Tan) An extension problem for the CR fractional Laplacian. Adv. Math. 270 (2015), 97 - 137. Article - Preprint
  80. (with T. Ekholm and H. Kovarik) Weak perturbations of the p-Laplacian. Calc. Var. Partial Differential Equations 53 (2015), 781 - 801. Article - Preprint
  81. (with A. Pushnitski) Trace class conditions for functions of Schrödinger operators. Comm. Math. Phys. 335 (2015), 477 - 496. Article - Preprint
  82. (with J. Sabin) Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates. Amer. J. of Math. 139 (2017), no. 6, 1649 - 1691. Article - Preprint
  83. (with E. A. Carlen and E. H. Lieb) Some operator and trace function convexity theorems. Linear Algebra Appl. 490 (2016), 174 - 185. Article - Preprint
  84. (with A. Pushnitski) The spectral density of a product of spectral projections. J. Funct. Anal. 268 (2015), 3867 - 3894. Article - Preprint
  85. (with R. D. Benguria and E. H. Lieb) Ground state energy of large polaron systems. J. Math. Phys. 56 (2015), 021901. Article - Preprint
  86. (with C. Hainzl, R. Seiringer and J. P. Solovej) The external field dependence of the BCS critical temperature. Comm. Math. Phys. 342 (2016), no. 1, 189 - 216. Article - Preprint
  87. (with E. H. Lieb) A compactness lemma and its application to the existence of minimizers for the liquid drop model. SIAM J. Math. Anal. 47 (2015), no. 6, 4436 - 4450. Article - Preprint
  88. (with B. Simon) Eigenvalue bounds for Schrödinger operators with complex potentials. II. J. Spectr. Theory. 7 (2017), no. 3, 633 - 658. Article - Preprint
  89. (with C. Hainzl, B. Schlein and R. Seiringer) Time-dependent Bogolubov-de-Gennes equations and non-validity of Ginzburg-Landau theory. Lett. Math. Phys. 106 (2016), no. 7, 913 - 923. Article - Preprint
  90. (with M. Lemm) Ginzburg-Landau theory with multiple order parameters: microscopic derivation and examples. Ann. H. Poincaré 17 (2016), no. 9, 2285 - 2340. Article - Preprint
  91. (with Z. Gang) Derivation of an effective evolution equation for a strongly coupled polaron. Anal. PDE 10 (2017), no. 2, 379 - 422. Article - Preprint
  92. Eigenvalue bounds for Schrödinger operators with complex potentials. III. Trans. Amer. Math. Soc. 370 (2018), no. 1, 219 - 240. Article - Preprint
  93. Eigenvalues of Schrödinger operators with complex surface potentials. In: Functional Analysis and Operator Theory for Quantum Physics, J. Dittrich, et al. (eds.), 245 - 260. EMS, 2017. Article - Preprint
  94. (with A. Laptev and O. Safronov) On the number of eigenvalues of Schrödinger operators with complex potentials. J. Lond. Math. Soc. (2) 94 (2016), no. 2, 377 - 390. Article - Preprint
  95. (with J. Behrndt, C. Kühn, V. Lotoreichik and J. Rohleder) Spectral theory for Schrödinger operators with δ-interactions supported on curves in R3. Ann. H. Poincaré 18 (2017), no. 4, 1305 - 1347. Article - Preprint
  96. (with E. H. Lieb and J. Sabin) Maximizers for the Stein-Tomas inequality. Geom. Funct. Anal. 26 (2016), no. 4, 1095 - 1134. Article - Preprint
  97. Eigenvalue bounds for the fractional Laplacian: A review. In: Recent Developments in Nonlocal Theory, G. Palatucci, T. Kuusi (eds.), 210 - 235, De Gruyter, 2017. Article - Preprint
  98. (with R. Killip and P. T. Nam) Nonexistence of large nuclei in the liquid drop model. Lett. Math. Phys. 106 (2016), no. 8, 1033 - 1036. Article - Preprint
  99. (with P. T. Nam and H. Van Den Bosch) The ionization conjecture in Thomas-Fermi-Dirac-von Weizsäcker theory. Comm. Pure Appl. Math. 71 (2018), no. 3, 577 - 614. Article - Preprint
  100. (with E. H. Lieb) A `liquid-solid' phase transition in a simple model for swarming, based on the `No Flat-Spots' theorem for subharmonic functions. Indiana Univ. Math. J. 67 (2018), no. 4, 1547 - 1569. Article - Preprint
  101. (with M. Lemm and B. Simon) Condensation of fermion pairs in a domain. Calc. Var. Partial Differential Equations 56 (2017), no. 2, 54. Article - Preprint
  102. (with P. T. Nam and H. Van Den Bosch) The maximal excess charge in Müller density-matrix-functional theory. Ann. H. Poincaré 19 (2018), no. 9, 2839 - 2867. Article - Preprint
  103. (with J. Sabin) Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces. Adv. Math. 317 (2017), 157 - 192. Article - Preprint
  104. (with J. Sabin) The Stein-Tomas inequality in trace ideals. Séminaire Laurent Schwartz - Équations aux dérivées partielles et applications. Année 2015 - 2016, Exp. No. XV, 12 pp., Ed. Éc. Polytech., Palaiseau, 2016. Article - Preprint
  105. (with L. Schimmer) Endpoint resolvent estimates for compact Riemannian manifolds. J. Funct. Anal. 272 (2017), no. 9, 3904 - 3918. Article - Preprint
  106. (with F. Méhats and C. Sparber) Averaging of nonlinear Schrödinger equations with strong magnetic confinement. Comm. Math. Sci. 15 (2017), no. 7, 1933 - 1945. Article - Preprint
  107. (with P. T. Nam and H. Van Den Bosch) A short proof of the ionization conjecture in Müller theory. In: Mathematical Problems in Quantum Physics, F. Bonetto, et al. (eds.), 1 - 12, Contemporary Mathematics 717, Amer. Math. Soc., 2018. Article - Preprint
  108. (with E. H. Lieb) Norms of quantum Gaussian multi-mode channels. J. Math. Phys. 58 (2017), 062204. Article - Preprint
  109. (with C. Hainzl and E. Langmann) The BCS critical temperature in a weak homogeneous magnetic field. J. Spectr. Theory 9 (2019), no. 3, 1005 - 1062. Article - Preprint
  110. (with T. Jin and J. Xiong) Minimizers for the fractional Sobolev inequality on domains. Calc. Var. Partial Differential Equations 57 (2018), no. 2, 57 - 43. Article - Preprint
  111. (with T. König) Classification of positive solutions to a nonlinear biharmonic equation with critical exponent. Anal. PDE 12 (2019), no. 4, 1101 - 1113. Article - Preprint
  112. (with A. Laptev) Bound on the number of negative eigenvalues of two-dimensional Schrödinger operators on domains. Algebra i Analiz 30 (2018), no. 3, 250 - 272. Reprinted in St. Petersburg Math. J. 30 (2019), no. 3, 573 - 589. Article - Preprint
  113. (with J. Sabin) Extremizers for the Airy-Strichartz inequality. Math. Ann. 372 (2018), no. 3-4, 1121 - 1166. Article - Preprint
  114. (with C. Hainzl) The BCS critical temperature in a weak external electric field via a linear two-body operator. In: D. Cadamuro, M. Duell, W. Dybalski, S. Simonella (eds), Macroscopic Limits of Quantum Systems. MaLiQS 2017. Springer Proceedings in Mathematics & Statistics, vol. 270. Springer, Cham. Article - Preprint
  115. (with J. A. Carrillo, M. G. Delgadino, J. Dolbeault and F. Hoffmann) Reverse Hardy-Littlewood-Sobolev inequalities. J. Math. Pures Appl. (9) 132 (2019), 133 - 165. Article - Preprint, see also Preprint
  116. (with E. A. Carlen and E. H. Lieb) Inequalities for quantum divergences and the Audenaert-Datta conjecture. J. Phys. A 51 (2018), no. 48, 483001. Article - Preprint
  117. (with E. A. Carlen, P. Ivanisvili and E. H. Lieb) Inequalities for Lp-norms that sharpen the triangle inequality and complement Hanner's inequality. J. Geom. Anal. 31 (2021), no. 4, 4051 - 4073. Article - Preprint
  118. (with K. Merz and H. Siedentop) Equivalence of Sobolev norms involving generalized Hardy operators. Int. Math. Res. Not. 3 (2021), 2284 - 2303. Article - Preprint
  119. (with L. Emmert and T. König) Liquid drop model for nuclear matter in the dilute limit. SIAM J. Math. Anal. 52 (2020), no. 2, 1980 - 1999. Article - Preprint
  120. (with E. H. Lieb) Periodic energy minimizers for a one-dimensional liquid drop model. Lett. Math. Phys. 109 (2019), no. 9, 2069 - 2081. Article - Preprint
  121. (with D. Hundertmark, M. Jex and P. T. Nam) The Lieb-Thirring inequality revisited. J. Eur. Math. Soc., to appear. Preprint
  122. (with A. Borichev and A. Volberg) Counting eigenvalues of Schrödinger operator with complex fast decreasing potential. Submitted. Preprint
  123. (with W. Borrelli) Sharp decay estimates for critical Dirac equations. Trans. Amer. Math. Soc. 373 (2020), no. 3, 2045 - 2070. Article - Preprint
  124. (with A. Pushnitski) Kato smoothness and functions of perturbed self-adjoint operators. Adv. Math. 351 (2019), 343 - 387. Article - Preprint
  125. (with A. Pushnitski) Schatten class conditions for functions of Schrödinger operators. Ann. Henri Poincaré 20 (2019), no. 11, 3543 - 3562. Article - Preprint
  126. (with S. Larson) Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain. J. Reine Angew. Math. 766 (2020), 195 - 228. Article - Preprint
  127. (with R. Seiringer) Quantum corrections to the Pekar asymptotics of a strongly coupled polaron. Comm. Pure Appl. Math. 74 (2021), no. 3, 544 - 588. Article - Preprint
  128. (with E. A. Carlen and E. H. Lieb) Inequalities that sharpen the triangle inequality for sums of N functions in Lp. Ark. Mat. 58 (2020), no. 1, 57 - 69. Article - Preprint
  129. (with T. König) Singular solutions to a semilinear biharmonic equation with a general critical nonlinearity. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), no. 4, 817 - 846. Article - Preprint
  130. Non-spherical equilibrium shapes in the liquid drop model. J. Math. Phys. 60 (2019), 071506. Article - Preprint
  131. (with Z. Gang) A non-linear adiabatic theorem for the one-dimensional Landau-Pekar equations. J. Funct. Anal. 279 (2020), no. 7, 108631. Article - Preprint
  132. (with K. Merz, H. Siedentop and B. Simon) Proof of the strong Scott conjecture for Chandrasekhar atoms. Pure and Applied Functional Analysis 5 (2020), no. 6, 1319 - 1356. Article - Preprint
  133. (with T. König and H. Kovarik) Energy asymptotics in the three-dimensional Brezis-Nirenberg problem. Calc. Var. Partial Differential Equations 60 (2021), Article number 58. Article - Preprint
  134. (with S. Kim and A. Pistoia) Non-degeneracy for the critical Lane-Emden system. Proc. Amer. Math. Soc. 149 (2021), no. 1, 265 - 278. Article - Preprint
  135. (with E. H. Lieb) Proof of spherical flocking based on quantitative rearrangement inequalities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), to appear. Preprint; see also the supplementary note
  136. (with T. König and H. Kovarik) Energy asymptotics in the Brezis-Nirenberg problem. The higher-dimensional case. Mathematics in Engineering 2 (2020), no. 1, 119 - 140. Preprint
  137. (with S. Larson) On the error in the two-term Weyl formula for the Dirichlet Laplacian. J. Math. Phys. 61 (2020), 043504. Article - Preprint
  138. (with D. Gontier and M. Lewin) The nonlinear Schrödinger equation for orthonormal functions: II. Application to Lieb-Thirring inequalities. Comm. Math. Phys., to appear. Preprint
  139. The Lieb-Thirring inequalities: Recent results and open problems. In: Linde Hall Inaugural Math Symposium, A. Kechris, et al. (eds.), Proceedings of Symposia in Pure Mathematics, Amer. Math. Soc., to appear. Preprint
  140. (with T. König and H. Tang) Classification of solutions of an equation related to a conformal log Sobolev inequality. Adv. Math. 375 (2020), 107395. Article - Preprint
  141. (with J. Sabin) Sharp Weyl laws with singular potentials. Submitted. Preprint
  142. (with K. Merz and H. Siedentop) Relativistic strong Scott conjecture: a short proof. Submitted. Preprint
  143. (with D. Gontier and M. Lewin) The periodic Lieb-Thirring inequality. In: Partial Differential Equations, Spectral Theory, and Mathematical Physics. The Ari Laptev Anniversary Volume, P. Exner, et al. (eds.). EMS Publishing House, to appear. Preprint
  144. (with S. Larson) Semiclassical asymptotics for a class of singular Schrödinger operators. In: Partial Differential Equations, Spectral Theory, and Mathematical Physics. The Ari Laptev Anniversary Volume, P. Exner, et al. (eds.). EMS Publishing House, to appear. Preprint
  145. (with H. Chen and H. Siedentop) A statistical theory of heavy atoms: energy and excess charge. Submitted. Preprint
  146. (with S. Larson) Two consequences of Davies's Hardy inequality. Functional Analysis and Its Applications, to appear. Preprint
  147. (with J. A. Carrillo, M. G. Delgadino and M. Lewin) Fast diffusion leads to partial mass concentration in Keller-Segel type stationary solutions. Submitted. Preprint
  148. (with M. Loss) Which magnetic fields support a zero mode? Submitted. Preprint
  149. (with P. T. Nam) Existence and nonexistence in the liquid drop model. Submitted. Preprint
  150. (with P. Ivanisvili) Hypercontractivity of the semigroup of the fractional laplacian on the n-sphere. Submitted. Preprint
  151. (with T. König and H. Kovarik) Blow-up of solutions of critical elliptic equation in three dimensions. Submitted. Preprint