<h1>Frank, Rupert</h1> <h2>Article from <a href="https://authors.library.caltech.edu">CaltechAUTHORS</a></h2> <ul> <li>Duong, Giao Ky and Frank, Rupert L., el al. (2024) <a href="https://authors.library.caltech.edu/records/evnx5-paj70">Cwikel–Lieb–Rozenblum type inequalities for Hardy–Schrödinger operator</a>; Journal de Mathématiques Pures et Appliquées; Vol. 190; 103598; <a href="https://doi.org/10.1016/j.matpur.2024.103598">10.1016/j.matpur.2024.103598</a></li> <li>Frank, Rupert L. and Sukochev, Fedor, el al. (2024) <a href="https://authors.library.caltech.edu/records/7sz4q-0za84">Endpoint Schatten class properties of commutators</a>; Advances in Mathematics; Vol. 450; 109738; <a href="https://doi.org/10.1016/j.aim.2024.109738">10.1016/j.aim.2024.109738</a></li> <li>Frank, Rupert L. and Hoffmann-Ostenhof, Thomas, el al. (2024) <a href="https://authors.library.caltech.edu/records/cvhrx-tay03">Hardy inequalities for large fermionic systems</a>; Journal of Spectral Theory; Vol. 14; No. 2; 805-835; <a href="https://doi.org/10.4171/jst/511">10.4171/jst/511</a></li> <li>Frank, Rupert and Laptev, Ari, el al. (2024) <a href="https://authors.library.caltech.edu/records/4ttta-y1w22">Weighted CLR type bounds in two dimensions</a>; Transactions of the American Mathematical Society; Vol. 377; No. 5; 3357-3371; <a href="https://doi.org/10.1090/tran/9124">10.1090/tran/9124</a></li> <li>Frank, Rupert L. (2023) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20230525-771644200.5">Sharp inequalities for coherent states and their optimizers</a>; Advanced Nonlinear Studies; Vol. 23; Art. No. 20220050; <a href="https://doi.org/10.1515/ans-2022-0050">10.1515/ans-2022-0050</a></li> <li>Frank, Rupert and Sukochev, Fedor, el al. (2023) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20230124-11595100.7">Asymptotics of singular values for quantum derivatives</a>; Transactions of the American Mathematical Society; Vol. 376; No. 3; 2047-2088; <a href="https://doi.org/10.1090/tran/8827">10.1090/tran/8827</a></li> <li>Frank, Rupert L. and Merz, Konstantin, el al. (2023) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20230209-988069100.27">The Scott conjecture for large Coulomb systems: a review</a>; Letters in Mathematical Physics; Vol. 113; Art. No. 11; <a href="https://doi.org/10.1007/s11005-023-01631-9">10.1007/s11005-023-01631-9</a></li> <li>Frank, Rupert L. and Larson, Simon (2022) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20221019-344256700.16">An inequality for the normal derivative of the Lane-Emden ground state</a>; Advances in Calculus of Variations; <a href="https://doi.org/10.1515/acv-2022-0005">10.1515/acv-2022-0005</a></li> <li>Frank, Rupert L. (2022) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20230227-87934600.14">Degenerate stability of some Sobolev inequalities</a>; Analyse Nonlineaire, Annales Institute H. Poincaré; Vol. 39; No. 6; 1459-1484; <a href="https://doi.org/10.4171/aihpc/35">10.4171/aihpc/35</a></li> <li>Frank, Rupert L. and Loss, Michael (2022) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20211004-222705713">Which magnetic fields support a zero mode?</a>; Journal für die reine und angewandte Mathematik; <a href="https://doi.org/10.1515/crelle-2022-0015">10.1515/crelle-2022-0015</a></li> <li>Carrillo, J. A. and Delgadino, M. G., el al. (2022) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20211004-222702303">Fast Diffusion leads to partial mass concentration in Keller–Segel type stationary solutions</a>; Mathematical Models and Methods in Applied Sciences; Vol. 32; No. 4; 831-850; <a href="https://doi.org/10.1142/S021820252250018X">10.1142/S021820252250018X</a></li> <li>Borichev, Alexander and Frank, Rupert L., el al. (2022) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20211202-191328270">Counting eigenvalues of Schrödinger operators with fast decaying complex potentials</a>; Advances in Mathematics; Vol. 397; Art. No. 108115; <a href="https://doi.org/10.1016/j.aim.2021.108115">10.1016/j.aim.2021.108115</a></li> <li>Frank, Rupert L. (2022) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20211208-559893000">Minimizers for a one-dimensional interaction energy</a>; Nonlinear Analysis; Vol. 216; Art. No. 112691; <a href="https://doi.org/10.1016/j.na.2021.112691">10.1016/j.na.2021.112691</a></li> <li>Frank, Rupert L. and König, Tobias, el al. (2022) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20211004-232831449">Reverse conformally invariant Sobolev inequalities on the sphere</a>; Journal of Functional Analysis; Vol. 282; No. 4; Art. No. 109339; <a href="https://doi.org/10.1016/j.jfa.2021.109339">10.1016/j.jfa.2021.109339</a></li> <li>Dolbeault, Jean and Frank, Rupert L., el al. (2022) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20211018-185208122">Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system</a>; Comptes Rendus Mathématique; Vol. 359; No. 10; 1279-1293; <a href="https://doi.org/10.5802/crmath.272">10.5802/crmath.272</a></li> <li>Frank, Rupert L. and Nam, Phan Thành (2021) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20211011-213336774">Existence and nonexistence in the liquid drop model</a>; Calculus of Variations and Partial Differential Equations; Vol. 60; No. 6; Art. No. 223; <a href="https://doi.org/10.1007/s00526-021-02072-9">10.1007/s00526-021-02072-9</a></li> <li>Frank, Rupert L. and Ivanisvili, Paata (2021) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20210628-191053213">Hypercontractivity of the semigroup of the fractional Laplacian on the n-sphere</a>; Journal of Functional Analysis; Vol. 281; No. 8; Art. No. 109145; <a href="https://doi.org/10.1016/j.jfa.2021.109145">10.1016/j.jfa.2021.109145</a></li> <li>Frank, Rupert L. and Lieb, Elliott H. (2021) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20211201-160010515">Proof of spherical flocking based on quantitative rearrangement inequalities</a>; Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V; Vol. 22; No. 3; 1241-1263; <a href="https://doi.org/10.2422/2036-2145.201909_007">10.2422/2036-2145.201909_007</a></li> <li>Frank, Rupert L. and Hundertmark, Dirk, el al. (2021) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20210713-164447574">The Lieb-Thirring inequality revisited</a>; Journal of the European Mathematical Society; Vol. 23; No. 8; 2583-2600; <a href="https://doi.org/10.4171/jems/1062">10.4171/jems/1062</a></li> <li>Frank, Rupert L. and Gontier, David, el al. (2021) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20200819-152827513">The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities</a>; Communications in Mathematical Physics; Vol. 384; No. 3; 1783-1828; <a href="https://doi.org/10.1007/s00220-021-04039-5">10.1007/s00220-021-04039-5</a></li> <li>Frank, R. L. and Larson, S. (2021) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20210416-094618696">Two Consequences of Davies' Hardy Inequality</a>; Functional Analysis and Its Applications; Vol. 55; No. 2; 174-177; <a href="https://doi.org/10.1134/S0016266321020106">10.1134/S0016266321020106</a></li> <li>Frank, Rupert L. and König, Tobias, el al. (2021) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20210222-100903531">Energy asymptotics in the three-dimensional Brezis–Nirenberg problem</a>; Calculus of Variations and Partial Differential Equations; Vol. 60; No. 2; Art. No. 58; <a href="https://doi.org/10.1007/s00526-021-01929-3">10.1007/s00526-021-01929-3</a></li> <li>Carlen, Eric A. and Frank, Rupert L., el al. (2021) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20200526-144324896">Inequalities for Lᵖ-Norms that Sharpen the Triangle Inequality and Complement Hanner's Inequality</a>; Journal of Geometric Analysis; Vol. 31; No. 4; 4051-4073; <a href="https://doi.org/10.1007/s12220-020-00425-y">10.1007/s12220-020-00425-y</a></li> <li>Frank, Rupert L. and Seiringer, Robert (2021) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20200928-152152135">Quantum Corrections to the Pekar Asymptotics of a Strongly Coupled Polaron</a>; Communications on Pure and Applied Mathematics; Vol. 74; No. 3; 544-588; <a href="https://doi.org/10.1002/cpa.21944">10.1002/cpa.21944</a></li> <li>Frank, Rupert L. and Merz, Konstantin, el al. (2021) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20190520-135826551">Equivalence of Sobolev Norms Involving Generalized Hardy Operators</a>; International Mathematics Research Notices; Vol. 2021; No. 3; 2284-2303; <a href="https://doi.org/10.1093/imrn/rnz135">10.1093/imrn/rnz135</a></li> <li>Frank, Rupert L. and König, Tobias, el al. (2020) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20200910-141623696">Classification of solutions of an equation related to a conformal log Sobolev inequality</a>; Advances in Mathematics; Vol. 375; Art. No. 107395; <a href="https://doi.org/10.1016/j.aim.2020.107395">10.1016/j.aim.2020.107395</a></li> <li>Frank, Rupert L. and Kim, Seunghyeok, el al. (2020) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20210114-143038456">Non-degeneracy for the critical Lane–Emden system</a>; Proceedings of the American Mathematical Society; Vol. 149; No. 1; 265-278; <a href="https://doi.org/10.1090/proc/15217">10.1090/proc/15217</a></li> <li>Frank, Rupert L. and Gang, Zhou (2020) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20200515-091132161">A non-linear adiabatic theorem for the one-dimensional Landau–Pekar equations</a>; Journal of Functional Analysis; Vol. 279; No. 7; Art. No. 108631; <a href="https://doi.org/10.1016/j.jfa.2020.108631">10.1016/j.jfa.2020.108631</a></li> <li>Frank, Rupert L. and Larson, Simon (2020) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20190520-133710218">Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain</a>; Journal für die reine und angewandte Mathematik; Vol. 2020; No. 766; 195-228; <a href="https://doi.org/10.1515/crelle-2019-0019">10.1515/crelle-2019-0019</a></li> <li>Carlen, Eric A. and Frank, Rupert L., el al. (2020) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20190520-133118107">Inequalities that sharpen the triangle inequality for sums of N functions in L^p</a>; Arkiv för Matematik; Vol. 58; No. 1; 57-69; <a href="https://doi.org/10.4310/ARKIV.2020.v58.n1.a4">10.4310/ARKIV.2020.v58.n1.a4</a></li> <li>Emmert, Lukas and Frank, Rupert L., el al. (2020) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20190520-135621604">Liquid Drop Model for Nuclear Matter in the Dilute Limit</a>; SIAM Journal on Mathematical Analysis; Vol. 52; No. 2; 1980-1999; <a href="https://doi.org/10.1137/19M1274420">10.1137/19M1274420</a></li> <li>Frank, Rupert L. and Larson, Simon (2020) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20200420-145908047">On the error in the two-term Weyl formula for the Dirichlet Laplacian</a>; Journal of Mathematical Physics; Vol. 61; No. 4; Art. No. 043504; <a href="https://doi.org/10.1063/1.5145003">10.1063/1.5145003</a></li> <li>Borrelli, William and Frank, Rupert L. (2020) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20190520-135426746">Sharp decay estimates for critical Dirac equations</a>; Transactions of the American Mathematical Society; Vol. 373; No. 3; 2045-2070; <a href="https://doi.org/10.1090/tran/7958">10.1090/tran/7958</a></li> <li>Frank, Rupert L. and Merz, Konstantin, el al. (2020) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20200109-110319953">Proof of the Strong Scott Conjecture for Chandrasekhar Atoms</a>; Pure and Applied Functional Analysis; Vol. 5; No. 6; 1319-1356; <a href="https://doi.org/10.48550/arXiv.1907.04894">10.48550/arXiv.1907.04894</a></li> <li>Carrillo, José A. and Delgadino, Matías G., el al. (2019) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20180604-111058055">Reverse Hardy-Littlewood-Sobolev inequalities</a>; Journal de Mathématiques Pures et Appliquées; Vol. 132; 133-165; <a href="https://doi.org/10.1016/j.matpur.2019.09.001">10.1016/j.matpur.2019.09.001</a></li> <li>Frank, Rupert L. and König, Tobias (2019) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20190520-133342030">Singular solutions to a semilinear biharmonic equation with a general critical nonlinearity</a>; Rendiconti Lincei - Matematica e Applicazioni; Vol. 30; No. 4; 817-846; <a href="https://doi.org/10.4171/RLM/871">10.4171/RLM/871</a></li> <li>Frank, Rupert L. and Pushnitski, Alexander (2019) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20190520-133945892">Schatten class conditions for functions of Schrödinger operators</a>; Annales Henri Poincaré; Vol. 20; No. 11; 3543-3562; <a href="https://doi.org/10.1007/s00023-019-00838-8">10.1007/s00023-019-00838-8</a></li> <li>Frank, Rupert L. and Lieb, Elliott H. (2019) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20190429-123953827">Periodic energy minimizers for a one-dimensional liquid drop model</a>; Letters in Mathematical Physics; Vol. 109; No. 9; 2069-2081; <a href="https://doi.org/10.1007/s11005-019-01171-1">10.1007/s11005-019-01171-1</a></li> <li>Frank, Rupert L. and Pushnitski, Alexander (2019) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20190520-102006314">Kato smoothness and functions of perturbed self-adjoint operators</a>; Advances in Mathematics; Vol. 351; 343-387; <a href="https://doi.org/10.1016/j.aim.2019.05.002">10.1016/j.aim.2019.05.002</a></li> <li>Frank, Rupert L. (2019) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20190520-132849818">Non-spherical equilibrium shapes in the liquid drop model</a>; Journal of Mathematical Physics; Vol. 60; No. 7; Art. No. 071506; <a href="https://doi.org/10.1063/1.5095603">10.1063/1.5095603</a></li> <li>Frank, R. L. and Laptev, A. (2019) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20180604-111721567">Bound on the number of negative eigenvalues of two-dimensional Schrödinger operators on domains</a>; ISBN 978-3-03719-175-0; Functional Analysis and Operator Theory for Quantum Physics: The Pavel Exner Anniversary Volume; St. Petersburg Mathematical Journal; Vol. 30; No. 3; 573-589; <a href="https://doi.org/10.1090/spmj/1559">10.1090/spmj/1559</a></li> <li>Frank, Rupert L. and König, Tobias (2019) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20180604-112112515">Classification of positive singular solutions to a nonlinear biharmonic equation with critical exponent</a>; Analysis & PDE; Vol. 12; No. 4; 1101-1113; <a href="https://doi.org/10.2140/apde.2019.12.1101">10.2140/apde.2019.12.1101</a></li> <li>Frank, Rupert L. and Hainzl, Christian, el al. (2019) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20180604-112611120">The BCS critical temperature in a weak homogeneous magnetic field</a>; Journal of Spectral Theory; Vol. 9; No. 3; 1005-1062; <a href="https://doi.org/10.4171/JST/270">10.4171/JST/270</a></li> <li>Frank, Rupert L. and Sabin, Julien (2018) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20180604-091720191">Extremizers for the Airy–Strichartz inequality</a>; Mathematische Annalen; Vol. 372; No. 3-4; 1121-1166; <a href="https://doi.org/10.1007/s00208-018-1695-7">10.1007/s00208-018-1695-7</a></li> <li>Carlen, Eric A. and Frank, Rupert L., el al. (2018) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20181106-123858044">Inequalities for quantum divergences and the Audenaert–Datta conjecture</a>; Journal of Physics A: Mathematical and Theoretical; Vol. 51; No. 48; Art. No. 483001; <a href="https://doi.org/10.1088/1751-8121/aae8a3">10.1088/1751-8121/aae8a3</a></li> <li>Frank, Rupert L. and Nam, Phan Thành, el al. (2018) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170428-155623144">The Maximal Excess Charge in Müller Density-Matrix-Functional Theory</a>; Annales Henri Poincaré; Vol. 19; No. 9; 2839-2867; <a href="https://doi.org/10.1007/s00023-018-0695-1">10.1007/s00023-018-0695-1</a></li> <li>Frank, Rupert L. and Jin, Tianling, el al. (2018) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20180221-110808137">Minimizers for the fractional Sobolev inequality on domains</a>; Calculus of Variations and Partial Differential Equations; Vol. 57; No. 2; Art. No. 43; <a href="https://doi.org/10.1007/s00526-018-1304-3">10.1007/s00526-018-1304-3</a></li> <li>Frank, Rupert L. and Nam, Phan Thành, el al. (2018) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-083145482">The ionization conjecture in Thomas-Fermi-Dirac-von Weizsäcker theory</a>; Communications on Pure and Applied Mathematics; Vol. 71; No. 3; 577-614; <a href="https://doi.org/10.1002/cpa.21717">10.1002/cpa.21717</a></li> <li>Frank, Rupert L. and Lieb, Elliott H. (2018) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20180912-155642789">A "Liquid-Solid" Phase Transition in a Simple Model for Swarming, Based on the "No Flat-Spots" Theorem for Subharmonic Functions</a>; Indiana University Mathematics Journal; Vol. 67; No. 4; 1547-1569; <a href="https://doi.org/10.1512/iumj.2018.67.7398">10.1512/iumj.2018.67.7398</a></li> <li>Frank, Rupert L. and Sabin, Julien (2017) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-160524071">Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates</a>; American Journal of Mathematics; Vol. 139; No. 6; 1649-1691; <a href="https://doi.org/10.1353/ajm.2017.0041">10.1353/ajm.2017.0041</a></li> <li>Frank, Rupert L. and Simon, Barry (2017) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170502-082840587">Eigenvalue bounds for Schrödinger operators with complex potentials. II</a>; Journal of Spectral Theory; Vol. 7; No. 3; 633-658; <a href="https://doi.org/10.4171/JST/173">10.4171/JST/173</a></li> <li>Frank, Rupert L. and Sabin, Julien (2017) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170428-155217715">Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces</a>; Advances in Mathematics; Vol. 317; 157-192; <a href="https://doi.org/10.1016/j.aim.2017.06.023">10.1016/j.aim.2017.06.023</a></li> <li>Frank, Rupert L. (2017) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170502-081943954">Eigenvalue bounds for Schrödinger operators with complex potentials. III</a>; Transactions of the American Mathematical Society; Vol. 370; 219-240; <a href="https://doi.org/10.1090/tran/6936">10.1090/tran/6936</a></li> <li>Frank, Rupert L. and Lieb, Elliott H. (2017) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170720-090957243">Norms of quantum Gaussian multi-mode channels</a>; Journal of Mathematical Physics; Vol. 58; No. 6; Art. No. 062204; <a href="https://doi.org/10.1063/1.4989809">10.1063/1.4989809</a></li> <li>Frank, Rupert L. and Schimmer, Lukas (2017) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170427-140719939">Endpoint resolvent estimates for compact Riemannian manifolds</a>; Journal of Functional Analysis; Vol. 272; No. 9; 3904-3918; <a href="https://doi.org/10.1016/j.jfa.2016.11.012">10.1016/j.jfa.2016.11.012</a></li> <li>Frank, Rupert L. and Lemm, Marius, el al. (2017) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170428-160716872">Condensation of fermion pairs in a domain</a>; Calculus of Variations and Partial Differential Equations; Vol. 56; Art. No. 54; <a href="https://doi.org/10.1007/s00526-017-1140-x">10.1007/s00526-017-1140-x</a></li> <li>Behrndt, Jussi and Frank, Rupert L., el al. (2017) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170417-074808881">Spectral Theory for Schrödinger Operators with δ-Interactions Supported on Curves in R³</a>; Annales Henri Poincaré; Vol. 18; No. 4; 1305-1347; <a href="https://doi.org/10.1007/s00023-016-0532-3">10.1007/s00023-016-0532-3</a></li> <li>Frank, Rupert L. and Méhats, Florian, el al. (2017) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170428-154120889">Averaging of nonlinear Schrödinger equations with strong magnetic confinement</a>; Communications in Mathematical Sciences; Vol. 15; No. 7; 1933-1945; <a href="https://doi.org/10.4310/CMS.2017.v15.n7.a7">10.4310/CMS.2017.v15.n7.a7</a></li> <li>Frank, Rupert L. and Zhou, Gang (2017) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170427-075023929">Derivation of an effective evolution equation for a strongly coupled polaron</a>; Analysis & PDE; Vol. 10; No. 2; 379-422; <a href="https://doi.org/10.2140/apde.2017.10.379">10.2140/apde.2017.10.379</a></li> <li>Frank, Rupert L. and Laptev, Ari, el al. (2016) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20161128-153447509">On the number of eigenvalues of Schrödinger operators with complex potentials</a>; Journal of the London Mathematical Society; Vol. 94; No. 2; 377-390; <a href="https://doi.org/10.1112/jlms/jdw039">10.1112/jlms/jdw039</a></li> <li>Frank, Rupert L. and Lenzmann, Enno, el al. (2016) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20160811-074144777">Uniqueness of Radial Solutions for the Fractional Laplacian</a>; Communications on Pure and Applied Mathematics; Vol. 69; No. 9; 1671-1726; <a href="https://doi.org/10.1002/cpa.21591">10.1002/cpa.21591</a></li> <li>Frank, Rupert L. and Lemm, Marius (2016) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20160930-133917913">Multi-Component Ginzburg-Landau Theory: Microscopic Derivation and Examples</a>; Annales Henri Poincaré; Vol. 17; No. 9; 2285-2340; <a href="https://doi.org/10.1007/s00023-016-0473-x">10.1007/s00023-016-0473-x</a></li> <li>Frank, Rupert L. and Killip, Rowan, el al. (2016) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20160805-093315174">Nonexistence of Large Nuclei in the Liquid Drop Model</a>; Letters in Mathematical Physics; Vol. 106; No. 8; 1033-1036; <a href="https://doi.org/10.1007/s11005-016-0860-8">10.1007/s11005-016-0860-8</a></li> <li>Frank, Rupert L. and Hainzl, Christian, el al. (2016) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20160722-140945826">Incompatibility of Time-Dependent Bogoliubov–de-Gennes and Ginzburg–Landau Equations</a>; Letters in Mathematical Physics; Vol. 106; No. 7; 913-923; <a href="https://doi.org/10.1007/s11005-016-0847-5">10.1007/s11005-016-0847-5</a></li> <li>Frank, Rupert L. and Lieb, Elliott H., el al. (2016) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20161129-084514277">Maximizers for the Stein–Tomas Inequality</a>; Geometric and Functional Analysis; Vol. 26; No. 4; 1095-1134; <a href="https://doi.org/10.1007/s00039-016-0380-9">10.1007/s00039-016-0380-9</a></li> <li>Frank, Rupert L. and Geisinger, Leander (2016) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20160317-104639269">Refined semiclassical asymptotics for fractional powers of the Laplace operator</a>; Journal für die reine und angewandte Mathematik; Vol. 2016; No. 712; 1-37; <a href="https://doi.org/10.1515/crelle-2013-0120">10.1515/crelle-2013-0120</a></li> <li>Carlen, Eric A. and Frank, Rupert L., el al. (2016) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20160225-075610491">Some operator and trace function convexity theorems</a>; Linear Algebra and its Applications; Vol. 490; 174-185; <a href="https://doi.org/10.1016/j.laa.2015.11.006">10.1016/j.laa.2015.11.006</a></li> <li>Frank, Rupert L. and Hainzl, Christian, el al. (2016) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20160303-112827308">The External Field Dependence of the BCS Critical Temperature</a>; Communications in Mathematical Physics; Vol. 342; No. 1; 189-216; <a href="https://doi.org/10.1007/s00220-015-2526-2">10.1007/s00220-015-2526-2</a></li> <li>Frank, Rupert L. and Sabin, Julien (2016) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170428-154841451">The Stein-Tomas inequality in trace ideals</a>; Séminaire Laurent Schwartz - EDP et applications; Vol. 15; Art. No. 12; <a href="https://doi.org/10.5802/slsedp.92">10.5802/slsedp.92</a></li> <li>Frank, Rupert L. and Lieb, Elliott H. (2015) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20160115-123530660">A Compactness Lemma and Its Application to the Existence of Minimizers for the Liquid Drop Model</a>; SIAM Journal on Mathematical Analysis; Vol. 47; No. 6; 4436-4450; <a href="https://doi.org/10.1137/15M1010658">10.1137/15M1010658</a></li> <li>Frank, Rupert L. and Geisinger, Leander (2015) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20150706-132935949">The Ground State Energy of a Polaron in a Strong Magnetic Field</a>; Communications in Mathematical Physics; Vol. 338; No. 1; 1-29; <a href="https://doi.org/10.1007/s00220-015-2367-z">10.1007/s00220-015-2367-z</a></li> <li>Ekholm, Tomas and Frank, Rupert L., el al. (2015) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20150604-085020731">Weak perturbations of the p-Laplacian</a>; Calculus of Variations and Partial Differential Equations; Vol. 53; No. 3-4; 781-801; <a href="https://doi.org/10.1007/s00526-014-0767-0">10.1007/s00526-014-0767-0</a></li> <li>Frank, Rupert L. and Pushnitski, Alexander (2015) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20150612-085811844">The spectral density of a product of spectral projections</a>; Journal of Functional Analysis; Vol. 268; No. 12; 3867-3894; <a href="https://doi.org/10.1016/j.jfa.2015.03.018">10.1016/j.jfa.2015.03.018</a></li> <li>Frank, Rupert L. and Pushnitski, Alexander (2015) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20150330-070612488">Trace Class Conditions for Functions of Schrödinger Operators</a>; Communications in Mathematical Physics; Vol. 335; No. 1; 477-496; <a href="https://doi.org/10.1007/s00220-014-2205-8">10.1007/s00220-014-2205-8</a></li> <li>Bengurla, Rafael D. and Frank, Rupert L., el al. (2015) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20150409-135549549">Ground state energy of large polaron systems</a>; Journal of Mathematical Physics; Vol. 56; No. 2; Art. No. 021901; <a href="https://doi.org/10.1063/1.4908125">10.1063/1.4908125</a></li> <li>Frank, Rupert L. and González, María del Mar, el al. (2015) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20150226-131020302">An extension problem for the CR fractional Laplacian</a>; Advances in Mathematics; Vol. 270; 97-137; <a href="https://doi.org/10.1016/j.aim.2014.09.026">10.1016/j.aim.2014.09.026</a></li> <li>Bellazzini, Jacopo and Frank, Rupert L., el al. (2014) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20141211-085533969">Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems</a>; Mathematische Annalen; Vol. 360; No. 3-4; 653-673; <a href="https://doi.org/10.1007/s00208-014-1046-2">10.1007/s00208-014-1046-2</a></li> <li>Frank, Rupert L. and Schlein, Benjamin (2014) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20140708-142118299">Dynamics of a Strongly Coupled Polaron</a>; Letters in Mathematical Physics; Vol. 104; No. 8; 911-929; <a href="https://doi.org/10.1007/s11005-014-0700-7">10.1007/s11005-014-0700-7</a></li> <li>Frank, Rupert L. and Lewin, Mathieu, el al. (2014) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20141023-082135692">Strichartz inequality for orthonormal functions</a>; Journal of the European Mathematical Society; Vol. 16; No. 7; 1507-1526; <a href="https://doi.org/10.4171/JEMS/467">10.4171/JEMS/467</a></li> <li>Frank, Rupert L. and Lenz, Daniel, el al. (2014) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-080259012">Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory</a>; Journal of Functional Analysis; Vol. 266; No. 8; 4765-4808; <a href="https://doi.org/10.1016/j.jfa.2014.02.008">10.1016/j.jfa.2014.02.008</a></li> <li>Bellazzini, Jacopo and Frank, Rupert L., el al. (2014) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170504-103757590">Existence of ground states for negative ions at the binding threshold</a>; Reviews in Mathematical Physics; Vol. 26; No. 01; Art. No. 1350021; <a href="https://doi.org/10.1142/S0129055X13500219">10.1142/S0129055X13500219</a></li> <li>Carlen, Eric A. and Frank, Rupert L., el al. (2014) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20140422-141713585">Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator</a>; Geometric and Functional Analysis; Vol. 24; No. 1; 63-84; <a href="https://doi.org/10.1007/s00039-014-0253-z">10.1007/s00039-014-0253-z</a></li> <li>Frank, Rupert L. (2014) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20140530-072438434">Cwikel's theorem and the CLR inequality</a>; Journal of Spectral Theory; Vol. 4; No. 1; 1-21; <a href="https://doi.org/10.4171/JST/59">10.4171/JST/59</a></li> <li>Frank, Rupert L. and Lieb, Elliott H. (2013) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20140130-111730599">Monotonicity of a relative Rényi entropy</a>; Journal of Mathematical Physics; Vol. 54; No. 12; Art. No. 122201; <a href="https://doi.org/10.1063/1.4838835">10.1063/1.4838835</a></li> <li>Frank, Rupert L. and Lieb, Elliott H. (2013) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-112412599">Extended Quantum Conditional Entropy and Quantum Uncertainty Inequalities</a>; Communications in Mathematical Physics; Vol. 323; No. 2; 487-495; <a href="https://doi.org/10.1007/s00220-013-1775-1">10.1007/s00220-013-1775-1</a></li> <li>Frank, Rupert L. and Lenzmann, Enno (2013) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20130807-152723442">Uniqueness of non-linear ground states for fractional Laplacians in R</a>; Acta Mathematica; Vol. 210; No. 2; 261-318; <a href="https://doi.org/10.1007/s11511-013-0095-9">10.1007/s11511-013-0095-9</a></li> <li>Frank, Rupert L. and Kovařík, Hynek (2013) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20130815-102433930">Heat kernels of metric trees and applications</a>; SIAM Journal on Mathematical Analysis; Vol. 45; No. 3; 1027-1046; <a href="https://doi.org/10.1137/120886297">10.1137/120886297</a></li> <li>Frank, Rupert L. and Lieb, Elliott H., el al. (2013) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170502-144700862">Symmetry of Bipolaron Bound States for Small Coulomb Repulsion</a>; Communications in Mathematical Physics; Vol. 319; No. 2; 557-573; <a href="https://doi.org/10.1007/s00220-012-1604-y">10.1007/s00220-012-1604-y</a></li> <li>Frank, Rupert L. and Lewin, Mathieu, el al. (2013) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20130319-101032718">A positive density analogue of the Lieb–Thirring inequality</a>; Duke Mathematical Journal; Vol. 162; No. 3; 435-495; <a href="https://doi.org/10.1215/00127094-2019477">10.1215/00127094-2019477</a></li> <li>Frank, Rupert L. and Geisinger, Leander (2012) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170505-092151992">Semi-classical analysis of the Laplace operator with Robin boundary conditions</a>; Bulletin of Mathematical Sciences; Vol. 2; No. 2; 281-319; <a href="https://doi.org/10.1007/s13373-012-0028-5">10.1007/s13373-012-0028-5</a></li> <li>Frank, Rupert L. and Lieb, Elliott H. (2012) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-103316603">Entropy and the Uncertainty Principle</a>; Annales Henri Poincaré; Vol. 13; No. 8; 1711-1717; <a href="https://doi.org/10.1007/s00023-012-0175-y">10.1007/s00023-012-0175-y</a></li> <li>Frank, Rupert L. and Seiringer, Robert (2012) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-113527750">Lieb-Thirring inequality for a model of particles with point interactions</a>; Journal of Mathematical Physics; Vol. 53; No. 9; Art. No. 095201; <a href="https://doi.org/10.1063/1.3697416">10.1063/1.3697416</a></li> <li>Frank, Rupert L. and Hainzl, Christian, el al. (2012) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-090207067">Microscopic Derivation of Ginzburg-Landau Theory</a>; Journal of the American Mathematical Society; Vol. 25; No. 3; 667-713; <a href="https://doi.org/10.1090/S0894-0347-2012-00735-8">10.1090/S0894-0347-2012-00735-8</a></li> <li>Frank, Rupert L. and Lieb, Elliott H. (2012) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-064349426">Sharp constants in several inequalities on the Heisenberg group</a>; Annals of Mathematics; Vol. 176; No. 1; 349-381; <a href="https://doi.org/10.4007/annals.2012.176.1.6">10.4007/annals.2012.176.1.6</a></li> <li>Frank, Rupert L. and Loss, Michael (2012) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-085309159">Hardy-Sobolev-Maz'ya inequalities for arbitrary domains</a>; Journal de Mathématiques Pures et Appliquées; Vol. 97; No. 1; 39-54; <a href="https://doi.org/10.1016/j.matpur.2011.04.004">10.1016/j.matpur.2011.04.004</a></li> <li>Frank, Rupert L. and Olofsson, Rikard (2011) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-072120225">Eigenvalue bounds for Schrödinger operators with a homogeneous magnetic field</a>; Letters in Mathematical Physics; Vol. 97; 227-241; <a href="https://doi.org/10.1007/s11005-011-0499-4">10.1007/s11005-011-0499-4</a></li> <li>Frank, Rupert L. (2011) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-065723391">Eigenvalue bounds for Schrödinger operators with complex potentials</a>; Bulletin of the London Mathematical Society; Vol. 43; No. 4; 745-750; <a href="https://doi.org/10.1112/blms/bdr008">10.1112/blms/bdr008</a></li> <li>Frank, Rupert L. and Lieb, Elliott H., el al. (2011) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-071121790">Stability and Absence of Binding for Multi-Polaron Systems</a>; Publications mathématiques de l'IHÉS; Vol. 113; No. 1; 39-67; <a href="https://doi.org/10.1007/s10240-011-0031-5">10.1007/s10240-011-0031-5</a></li> <li>Frank, Rupert L. and Lewin, Mathieu, el al. (2011) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-093433748">Energy Cost to Make a Hole in the Fermi Sea</a>; Physical Review Letters; Vol. 106; No. 15; Art. No. 150402; <a href="https://doi.org/10.1103/PhysRevLett.106.150402">10.1103/PhysRevLett.106.150402</a></li> <li>Frank, Rupert L. and Simon, Barry (2011) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20110502-112300651">Critical Lieb-Thirring bounds in gaps and the generalized Nevai conjecture for finite gap Jacobi matrices</a>; Duke Mathematical Journal; Vol. 157; No. 3; 461-493; <a href="https://doi.org/10.1215/00127094-1272912">10.1215/00127094-1272912</a></li> <li>Ekholm, Tomas and Frank, Rupert L., el al. (2011) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170512-094610802">Eigenvalue estimates for Schrödinger operators on metric trees</a>; Advances in Mathematics; Vol. 226; No. 6; 5165-5197; <a href="https://doi.org/10.1016/j.aim.2011.01.001">10.1016/j.aim.2011.01.001</a></li> <li>Frank, Rupert L. and Morozov, Sergey, el al. (2011) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170526-094859466">Weakly coupled bound states of Pauli operators</a>; Calculus of Variations and Partial Differential Equations; Vol. 40; No. 1; 253-271; <a href="https://doi.org/10.1007/s00526-010-0339-x">10.1007/s00526-010-0339-x</a></li> <li>Frank, Rupert L. (2011) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-111716146">On the uniqueness of ground states of non-local equations</a>; Journées équations aux dérivées partielles; Vol. 2011; A5; <a href="https://doi.org/10.5802/jedp.77">10.5802/jedp.77</a></li> <li>Frank, Rupert L. and Lieb, Elliott H. (2010) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170510-143424777">Inversion positivity and the sharp Hardy–Littlewood–Sobolev inequality</a>; Calculus of Variations and Partial Differential Equations; Vol. 39; No. 1-2; 85-99; <a href="https://doi.org/10.1007/s00526-009-0302-x">10.1007/s00526-009-0302-x</a></li> <li>Frank, Rupert L. and Lieb, Elliott H., el al. (2010) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170501-064924933">Bipolaron and N-Polaron Binding Energies</a>; Physical Review Letters; Vol. 104; No. 21; Art. No. 210402; <a href="https://doi.org/10.1103/PhysRevLett.104.210402">10.1103/PhysRevLett.104.210402</a></li> <li>Frank, Rupert L. and Laptev, Ari (2010) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170526-092032067">Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group</a>; International Mathematics Research Notices; Vol. 2010; No. 15; 2889-2902; <a href="https://doi.org/10.1093/imrn/rnp230">10.1093/imrn/rnp230</a></li> <li>Frank, Rupert L. (2009) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170512-093621400">A Simple Proof of Hardy-Lieb-Thirring Inequalities</a>; Communications in Mathematical Physics; Vol. 290; No. 2; 789-800; <a href="https://doi.org/10.1007/s00220-009-0759-7">10.1007/s00220-009-0759-7</a></li> <li>Breuer, Jonathan and Frank, Rupert L. (2009) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20091012-105116089">Singular spectrum for radial trees</a>; Reviews in Mathematical Physics; Vol. 21; No. 7; 929-945; <a href="https://doi.org/10.1142/S0129055X09003773">10.1142/S0129055X09003773</a></li> <li>Frank, Rupert L. (2009) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170526-100006703">A note on low energy scattering for homogeneous long range potentials</a>; Annales Henri Poincaré; Vol. 10; No. 3; 573-575; <a href="https://doi.org/10.1007/s00023-009-0410-3">10.1007/s00023-009-0410-3</a></li> <li>Frank, Rupert L. and Seiringer, Robert (2008) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170510-155935658">Non-linear ground state representations and sharp Hardy inequalities</a>; Journal of Functional Analysis; Vol. 255; No. 12; 3407-3430; <a href="https://doi.org/10.1016/j.jfa.2008.05.015">10.1016/j.jfa.2008.05.015</a></li> <li>Frank, Rupert L. and Simon, Barry, el al. (2008) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170408-144423477">Eigenvalue Bounds for Perturbations of Schrödinger Operators and Jacobi Matrices With Regular Ground States</a>; Communications in Mathematical Physics; Vol. 282; No. 1; 199-208; <a href="https://doi.org/10.1007/s00220-008-0453-1">10.1007/s00220-008-0453-1</a></li> <li>Frank, Rupert L. and Siedentop, Heinz, el al. (2008) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170510-160420593">The Ground State Energy of Heavy Atoms: Relativistic Lowering of the Leading Energy Correction</a>; Communications in Mathematical Physics; Vol. 278; No. 2; 549-566; <a href="https://doi.org/10.1007/s00220-007-0397-x">10.1007/s00220-007-0397-x</a></li> <li>Ekholm, Tomas and Frank, Rupert (2008) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170531-151816388">Lieb–Thirring inequalities on the half-line with critical exponent</a>; European Mathematical Society; Vol. 10; No. 3; 739-755; <a href="https://doi.org/10.4171/JEMS/128">10.4171/JEMS/128</a></li> <li>Frank, Rupert L. and Hainzl, Christian, el al. (2007) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170510-105303048">The critical temperature for the BCS equation at weak coupling</a>; Journal of Geometric Analysis; Vol. 17; No. 4; 559-567; <a href="https://doi.org/10.1007/BF02937429">10.1007/BF02937429</a></li> <li>Frank, Rupert L. and Lieb, Elliott H., el al. (2007) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170510-095237526">Number of Bound States of Schröedinger Operators with Matrix-Valued Potentials</a>; Letters in Mathematical Physics; Vol. 82; No. 2-3; 107-116; <a href="https://doi.org/10.1007/s11005-007-0211-x">10.1007/s11005-007-0211-x</a></li> <li>Frank, Rupert L. and Laptev, Ari, el al. (2007) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170511-070749185">Eigenvalue estimates for magnetic Schrödinger operators in domains</a>; Proceedings of the American Mathematical Society; Vol. 136; No. 12; 4245-4255; <a href="https://doi.org/10.1090/S0002-9939-08-09523-3">10.1090/S0002-9939-08-09523-3</a></li> <li>Frank, Rupert L. and Lieb, Elliott H., el al. (2007) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170526-134159102">Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators</a>; Journal of the American Mathematical Society; Vol. 21; No. 4; 925-950; <a href="https://doi.org/10.1090/S0894-0347-07-00582-6">10.1090/S0894-0347-07-00582-6</a></li> <li>Frank, Rupert L. and Lieb, Elliott H., el al. (2007) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170601-073744902">Stability of Relativistic Matter with Magnetic Fields for Nuclear Charges up to the Critical Value</a>; Communications in Mathematical Physics; Vol. 275; No. 2; 479-489; <a href="https://doi.org/10.1007/s00220-007-0307-2">10.1007/s00220-007-0307-2</a></li> <li>Frank, Rupert L. (2007) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170601-081138045">On the asymptotic number of edge states for magnetic Schrödinger operators</a>; Proceedings of the London Mathematical Society; Vol. 95; No. 1; 1-19; <a href="https://doi.org/10.1112/plms/pdl024">10.1112/plms/pdl024</a></li> <li>Exner, Pavel and Frank, Rupert L. (2007) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170601-083108196">Absolute Continuity of the Spectrum for Periodically Modulated Leaky Wires in R^3</a>; Annales Henri Poincaré; Vol. 8; No. 2; 241-263; <a href="https://doi.org/10.1007/s00023-006-0307-3">10.1007/s00023-006-0307-3</a></li> <li>Frank, Rupert L. and Laptev, Ari, el al. (2006) <a href="https://resolver.caltech.edu/CaltechAUTHORS:20170601-075251570">Lieb–Thirring Inequalities for Schrödinger Operators with Complex-valued Potentials</a>; Letters in Mathematical Physics; Vol. 77; No. 3; 309-316; <a href="https://doi.org/10.1007/s11005-006-0095-1">10.1007/s11005-006-0095-1</a></li> </ul>