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We study the Harper-Hofstadter Hamiltonian and its corresponding non-perturbative butterfly spectrum.
1954
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P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field
1955
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D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields
1976
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E. Bogomolny, Calculation of instanton-anti-instanton contributions in quantum mechanics
1980
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J. Zinn-Justin, Multi-instanton contributions in quantum mechanics
1981
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M. Wilkinson, Critical properties of electron eigenstates in incommensurate systems
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S. Coleman, Aspects of symmetry: selected erice lectures
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D. Freed and J. A. Harvey, Instantons and the spectrum of bloch electrons in a magnetic field
1990
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Y. Hasegawa, Y. Hatsugai, M. Kohmoto, and G. Montambaux, Stabilization of flux states on two-dimensional lattices
1990
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J. Bellissard, C. Kreft, and R. Seiler, Analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods
1991
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M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes
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G. Álvarez and C. Casares, Exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator
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G. Álvarez and C. Casares, Uniform asymptotic and JWKB expansions for anharmonic oscillators
2000
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J. Zinn-Justin, Quantum field theory and critical phenomena
2002
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G. Álvarez, C. J. Howls, and H. J. Silverstone, Dispersive hyperasymptotics and the anharmonic oscillator
2002
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G. Álvarez, C. J. Howls, and H. J. Silverstone, Anharmonic oscillator discontinuity formulae up to second-exponentially-small order
2002
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J. Zinn-Justin and U. D. Jentschura, Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions
2004
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J. Zinn-Justin and U. D. Jentschura, Multi-instantons and exact results II: Specific cases, higher-order effects, and numerical calculations
2004
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G. Álvarez, Langer–Cherry derivation of the multi-instanton expansion for the symmetric double well
2004
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M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño, and C. Vafa, Topological strings and integrable hierarchies
2006
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G. V. Dunne, Functional determinants in quantum field theory
2008
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X. Wang, G. Zhang, and M.-x. Huang, New exact quantization condition for toric Calabi-Yau geometries
2015
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M. Mariño, Instantons and Large N
2015
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Y. Hatsuda, H. Katsura, and Y. Tachikawa, Hofstadter’s butterfly in quantum geometry
2016
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A. Grassi, Y. Hatsuda, and M. Marino, Topological strings from quantum mechanics
2016
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2016
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L. A. Takhtajan, Quantum Mechanics for Mathematicians
2008
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N. A. Nekrasov and S. L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories
2010
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2011
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2011
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2011
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M.-x. Huang and A. Klemm, Direct integration for general Ω \Omega backgrounds
2012
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2012
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J. Kallen and M. Mariño, Instanton effects and quantum spectral curves
2016
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A.-K. Kashani-Poor, Quantization condition from exact WKB for difference equations
2016
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2016
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M. Mariño and S. Zakany, Matrix models from operators and topological strings
2016
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Y. Hatsuda, Y. Sugimoto, and Z. Xu, Calabi-Yau geometry and electrons on 2d lattices
2017
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2017
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M. Mariño and S. Zakany, Exact eigenfunctions and the open topological string
2017
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G. Bonelli, A. Grassi, and A. Tanzini, Seiberg-Witten theory as a Fermi gas
2017
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J. N. Fuchs, F. Piéchon, and G. Montambaux, Landau levels, response functions and magnetic oscillations from a generalized onsager relation
2018
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N. Nekrasov, Tying up instantons with anti-instantons
2018
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