The Density-Potential Mapping in Quantum Dynamics “There and Back Again”
Total Page：16
File Type：pdf, Size：1020Kb
The Density-Potential Mapping in Quantum Dynamics “There and Back Again” Markus Penz Basic Research Community for Physics Dissertation submitted to the Faculty of Mathematics, Computer Science and Physics of the University of Innsbruck in partial fulfillment of the requirements for the degree of doctor of science. Advisor: Gebhard Gr¨ubl, University Innsbruck Secondary advisors: Michael Ruggenthaler, Max Planck Institute for the Struc- ture and Dynamics of Matter, and Robert van Leeuwen, University of Jyv¨askyl¨a Digital Edition Innsbruck, 18th October 2016 Abstract. This work studies in detail the possibility of defining a one-to-one mapping from charge densities as obtained by the time-dependent Schr¨odinger equation to external potentials. Such a mapping is provided by the Runge– Gross theorem and lies at the very core of time-dependent density functional theory. After introducing the necessary mathematical concepts, the usual map- ping “there”—from potentials to wave functions as solutions to the Schr¨odinger equation—is revisited paying special attention to Sobolev regularity. This is scrutinised further when the question of functional differentiability of the solu- tion with respect to the potential arises, a concept related to linear response theory. Finally, after a brief introduction to general density functional theory, the mapping “back again”—from densities to potentials thereby inverting the arXiv:1610.05552v1 [math-ph] 18 Oct 2016 Schr¨odingerequation for a fixed initial state—is defined. Apart from utilising the original Runge–Gross proof this is achieved through a fixed-point proce- dure. Both approaches give rise to mathematical issues, previously unresolved, which however could be dealt with to some extent within the framework at hand. c z To the extent possible under law, the author has waived all copyright and related or neighbouring rights to this work. It thereby belongs to public domain. This work is published from Austria. Contents 1 Easy Reading 1 1.1 Preamble . 1 1.2 Acknowledgements . 2 1.3 Epistemological disclaimer . 3 1.4 A note on non-analyticity . 6 1.5 The conception of molecules in quantum chemistry . 8 1.6 Remarks on notation and terminology . 11 2 Function Spaces and Duality 15 2.1 Sobolev spaces . 15 2.1.1 Classes of domains . 16 2.1.2 From Lebesgue to Sobolev spaces . 17 2.1.3 Sobolev embeddings . 18 2.1.4 Lipschitz and absolute continuity . 20 2.1.5 Zero boundary conditions . 20 2.1.6 Equivalence of Sobolev norms . 21 2.2 Unbounded theory of operators . 24 2.2.1 Preceding definitions . 25 2.2.2 The theorem of Hellinger–Toeplitz and closed operators . 26 2.2.3 Resolvent operators . 28 2.2.4 Positive operators . 29 2.2.5 Essentially self-adjoint operators and the Laplacian . 30 2.3 Dual spaces . 31 2.3.1 State–observable duality . 31 2.3.2 Lebesgue space dual . 33 2.3.3 Sobolev space dual . 33 2.3.4 The Lax–Milgram theorem . 34 3 Schr¨odinger Dynamics 40 3.1 The abstract Cauchy problem . 41 3.1.1 Evolution semigroups . 42 3.1.2 Types of solutions . 44 3.2 The Schr¨odingerinitial value problem . 46 3.2.1 Uniqueness of solutions . 47 i Contents ii 3.2.2 Conservation laws and energy spaces . 47 3.2.3 Spread of wave packets under free evolution . 48 3.2.4 Kovalevskaya and the Schr¨odingerequation . 50 3.3 Schr¨odingerdynamics with static Hamiltonians . 53 3.3.1 Bounded Hamiltonians (operator exponential) . 53 3.3.2 Unbounded Hamiltonians (Stone’s theorem) . 54 3.4 Classes of static potentials . 57 3.4.1 The Kato–Rellich theorem . 58 3.4.2 Kato perturbations . 61 3.4.3 Stummel class . 66 3.4.4 Kato class . 68 3.5 Properties of eigenfunctions . 69 3.5.1 Possible non-regularity of the associated density . 70 3.5.2 Unique continuation property of eigenstates . 71 3.5.3 Analyticity of eigenstates and densities . 73 3.6 Overview of results for time-dependent Hamiltonians . 74 3.7 The stepwise static approximation method . 77 3.7.1 Preparatory definitions and lemmas . 77 3.7.2 Existence of Schr¨odingersolutions . 82 3.8 Regularity in higher Sobolev norms . 85 3.8.1 Motivation from TDDFT . 86 3.8.2 On growth of Sobolev norms . 86 3.8.3 Sobolev regularity from the stepwise static approximation 87 3.9 The successive substitutions method . 91 3.9.1 Yajima’s set of Banach spaces . 91 3.9.2 It’s all about the inequalities . 93 3.9.3 The interaction picture and existence of Schr¨odingerso- lutions . 100 4 Functional Differentiability 105 4.1 Basics of variational calculus . 105 4.1.1 Gˆateauxand Fr´echet differentiability . 105 4.1.2 Results on equivalence of the two notions . 106 4.1.3 More results for Fr´echet differentiable maps . 108 4.1.4 The Landau symbols . 109 4.2 Variation of trajectories and observable quantities . 110 4.2.1 In the stepwise static approximation . 111 4.2.2 In the successive substitutions method . 113 4.2.3 Variation of bounded observable quantities . 118 4.2.4 Variation of unbounded observable quantities . 119 4.2.5 Variation of the density . 121 4.2.6 An inhomogeneous Schr¨odingerequation for δψ . 122 4.2.7 A relation to Duhamel’s principle . 124 4.2.8 Energy estimates from trajectory variations and another existence proof . 125 Contents iii 4.3 Linear response theory . 128 4.3.1 Smooth approximations to the density operator . 128 4.3.2 Deriving the density response . 129 4.3.3 Density response from the spectral measure of the Hamil- tonian . 130 4.3.4 Relation to the Lebesgue decomposition theorem . 131 4.3.5 The response function in frequency representation . 132 5 Introduction to Density Functional Theory 134 5.1 Precursors of DFT . 135 5.1.1 Motivation . 135 5.1.2 Thomas–Fermi theory . 136 5.1.3 Reduced density matrices . 137 5.1.4 Thomas–Fermi–Dirac theory and the local-density approx- imation . 140 5.2 Foundations of DFT . 143 5.2.1 The Hohenberg–Kohn theorem . 144 5.2.2 The Hohenberg–Kohn variation principle . 145 5.2.3 The self-consistent Hartree–Fock method . 147 5.2.4 The self-consistent Kohn–Sham equations . 148 5.3 Time-dependent DFT . 149 5.3.1 Basics of the Runge–Gross theorem . 150 5.3.2 The classical Runge–Gross proof . 152 5.3.3 The Kohn–Sham scheme and the extended Runge–Gross theorem . 153 5.3.4 Limitations of the Runge–Gross theorems . 157 5.3.5 Reformulation of the internal forces term . 158 5.3.6 Examination of the second law term . 162 5.3.7 Dipole laser-matter interaction as a special case . 163 6 A Fixed-Point Proof of the Runge–Gross Theorem 165 6.1 Weak solutions to a fixed-point scheme . 165 6.1.1 Construction of the fixed-point scheme . 165 6.1.2 Relation to a non-linear Schr¨odingerequation . 166 6.1.3 The dreaded ρ-problem . 167 6.1.4 Weak solutions to the Sturm–Liouville equation . 168 6.2 Potentials from weighted Sobolev spaces . 170 6.2.1 Construction of the Hilbert space of potentials . 170 6.2.2 The simple elliptic case . 171 6.2.3 Embedding theorems for weighted Sobolev spaces . 173 6.2.4 A proof of coercivity . 175 6.2.5 Application of the Lax–Milgram theorem . 176 6.2.6 Stronger solutions to the Sturm–Liouville equation . 178 6.3 Special cases . 180 6.3.1 The one-dimensional case . 180 6.3.2 The spherical-symmetric case . 181 Contents iv 6.3.3 The single particle case . 183 6.4 Towards a more rigorous Runge–Gross proof . 184 6.4.1 Transformation to a Schr¨odingerproblem . 184 6.4.2 Fr´echet estimate of the internal forces mapping . 184 6.4.3 Lipschitz estimate of the internal forces mapping . 186 6.4.4 More on fixed-point theorems . 187 6.4.5 Open issues . 188 6.4.6 Concluding remarks . 190 Bibliography 192 Chapter 1 Easy Reading Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth. — Marcus Aurelius, Meditations 1.1 Preamble The belief in the continuous progress of knowledge along with a steady or even exponential rise in economical wealth is contrasted by obvious limits to growth, outbreak of social struggles, or profound cultural changes, but still persists. Sci- ence especially is frequently thought as gradually improving our view towards an objective world, revealing what is true and providing us with ever expanding models to predict the future. This view is contrasted by the historic accounts of Fleck (1935) and Kuhn (1962) who collected evidence for revolutionary changes in the science world that created radical new thought collectives using com- pletely different concepts and language. That a body of knowledge is in itself not a linearly progressing set of statements also became apparent in the writing of this thesis. Parts that were once in the final chapter suddenly found their way to the far front just to be later moved to an appendix (that does not exist anymore) or utterly removed. It became obvious that a naturally built pre- sentation is hard to achieve, and the demanded form only allows for a treatise page by page. A more suitable format for the rhizomatic structure of knowledge (Deleuze–Guattari, 1987) without definite center, no full hierarchy, and many connections into all directions, which also marks the typical working progress, would be a hypertext document. Maybe this will find consideration in future academic standards.