Annales Henri Poincaré最新文献

{"title":"A Short Proof of Bose–Einstein Condensation in the Gross–Pitaevskii Regime and Beyond","authors":"Christian Brennecke,&nbsp;Morris Brooks,&nbsp;Cristina Caraci,&nbsp;Jakob Oldenburg","doi":"10.1007/s00023-024-01465-8","DOIUrl":"10.1007/s00023-024-01465-8","url":null,"abstract":"<div><p>We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order <span>( N^{kappa -1})</span>, for some <span>(kappa &gt;0)</span>. For the range <span>( kappa in [0, frac{1}{43}))</span>, Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for <span>( kappa in [0, frac{1}{20}))</span> by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"26 4","pages":"1353 - 1373"},"PeriodicalIF":1.4,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00023-024-01465-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spectral Convergence of the Dirac Operator on Typical Hyperbolic Surfaces of High Genus","authors":"Laura Monk,&nbsp;Rareş Stan","doi":"10.1007/s00023-024-01452-z","DOIUrl":"10.1007/s00023-024-01452-z","url":null,"abstract":"<div><p>In this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil–Petersson surfaces of large genus <i>g</i> with <span>(o(sqrt{g}))</span> cusps, we prove convergence of the spectral density to the spectral density of the hyperbolic plane, with quantitative error estimates. This result implies upper bounds on spectral counting functions and multiplicities, as well as a uniform Weyl law, true for typical hyperbolic surfaces equipped with any nontrivial spin structure.</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"26 1","pages":"365 - 387"},"PeriodicalIF":1.4,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00023-024-01452-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141532107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Time Functions on Lorentzian Length Spaces","authors":"Annegret Burtscher,&nbsp;Leonardo García-Heveling","doi":"10.1007/s00023-024-01461-y","DOIUrl":"10.1007/s00023-024-01461-y","url":null,"abstract":"<div><p>In general relativity, time functions are crucial objects whose existence and properties are intimately tied to the causal structure of a spacetime and also to the initial value formulation of the Einstein equations. In this work we establish all fundamental classical existence results on time functions in the setting of Lorentzian (pre-)length spaces (including causally plain continuous spacetimes, closed cone fields and even more singular spaces). More precisely, we characterize the existence of time functions by <i>K</i>-causality, show that a modified notion of Geroch’s volume functions are time functions if and only if the space is causally continuous, and lastly, characterize global hyperbolicity by the existence of Cauchy time functions, and Cauchy sets. Our results thus inevitably show that no manifold structure is needed in order to obtain suitable time functions.</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"26 5","pages":"1533 - 1572"},"PeriodicalIF":1.4,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00023-024-01461-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141532111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Fermionic Entanglement Entropy of the Vacuum State of a Schwarzschild Black Hole Horizon","authors":"Felix Finster,&nbsp;Magdalena Lottner","doi":"10.1007/s00023-024-01459-6","DOIUrl":"10.1007/s00023-024-01459-6","url":null,"abstract":"<div><p>We define and analyze the fermionic entanglement entropy of a Schwarzschild black hole horizon for the regularized vacuum state of an observer at infinity. Using separation of variables and an integral representation of the Dirac propagator, the entanglement entropy is computed to be a prefactor times the number of occupied angular momentum modes on the event horizon.\u0000</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"26 2","pages":"527 - 595"},"PeriodicalIF":1.4,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00023-024-01459-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Cauchy Problem for the Logarithmic Schrödinger Equation Revisited","authors":"Masayuki Hayashi,&nbsp;Tohru Ozawa","doi":"10.1007/s00023-024-01460-z","DOIUrl":"10.1007/s00023-024-01460-z","url":null,"abstract":"<div><p>We revisit the Cauchy problem for the logarithmic Schrödinger equation and construct strong solutions in <span>(H^1)</span>, the energy space, and the <span>(H^2)</span>-energy space. The solutions are provided in a constructive way, which does not rely on compactness arguments, that a sequence of approximate solutions forms a Cauchy sequence in a complete function space and then actual convergence is shown to be in a strong sense.</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"26 4","pages":"1209 - 1238"},"PeriodicalIF":1.4,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Simple Testbed for Stability Analysis of Quantum Dissipative Systems","authors":"Thierry Goudon,&nbsp;Simona Rota Nodari","doi":"10.1007/s00023-024-01458-7","DOIUrl":"10.1007/s00023-024-01458-7","url":null,"abstract":"<div><p>We study a two-state quantum system with a nonlinearity intended to describe interactions with a complex environment, arising through a nonlocal coupling term. We study the stability of particular solutions, obtained as constrained extrema of the energy functional of the system. The simplicity of the model allows us to justify a complete stability analysis. This is the opportunity to review in detail the techniques to investigate the stability issue. We also bring out the limitations of perturbative approaches based on simpler asymptotic models.</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"26 4","pages":"1149 - 1208"},"PeriodicalIF":1.4,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141532113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Essential Self-Adjointness of Even-Order, Strongly Singular, Homogeneous Half-Line Differential Operators","authors":"Fritz Gesztesy,&nbsp;Markus Hunziker,&nbsp;Gerald Teschl","doi":"10.1007/s00023-024-01451-0","DOIUrl":"10.1007/s00023-024-01451-0","url":null,"abstract":"<div><p>We consider essential self-adjointness on the space <span>(C_0^{infty }((0,infty )))</span> of even-order, strongly singular, homogeneous differential operators associated with differential expressions of the type </p><div><div><span>$$begin{aligned} tau _{2n}(c) = (-1)^n frac{d^{2n}}{d x^{2n}} + frac{c}{x^{2n}}, quad x &gt; 0, ; n in {{mathbb {N}}}, ; c in {{mathbb {R}}}, end{aligned}$$</span></div></div><p>in <span>(L^2((0,infty );dx))</span>. While the special case <span>(n=1)</span> is classical and it is well known that <span>(tau _2(c)big |_{C_0^{infty }((0,infty ))})</span> is essentially self-adjoint if and only if <span>(c ge 3/4)</span>, the case <span>(n in {{mathbb {N}}})</span>, <span>(n ge 2)</span>, is far from obvious. In particular, it is not at all clear from the outset that </p><div><div><span>$$begin{aligned} begin{aligned}&amp;textit{there exists }c_n in {{mathbb {R}}}, n in {{mathbb {N}}}textit{, such that} &amp;quad tau _{2n}(c)big |_{C_0^{infty }((0,infty ))} , textit{ is essentially self-adjoint}quad quad quad quad quad quad quad quad quad quad (*) {}&amp;quad textit{ if and only if } c ge c_n. end{aligned} end{aligned}$$</span></div></div><p>As one of the principal results of this paper we indeed establish the existence of <span>(c_n)</span>, satisfying <span>(c_n ge (4n-1)!!big /2^{2n})</span>, such that property (*) holds. In sharp contrast to the analogous lower semiboundedness question, </p><div><div><span>$$begin{aligned} textit{for which values of }ctextit{ is }tau _{2n}(c)big |_{C_0^{infty }((0,infty ))}{} textit{ bounded from below?}, end{aligned}$$</span></div></div><p>which permits the sharp (and explicit) answer <span>(c ge [(2n -1)!!]^{2}big /2^{2n})</span>, <span>(n in {{mathbb {N}}})</span>, the answer for (*) is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly, </p><div><div><span>$$begin{aligned} c_{1}&amp;= 3/4, quad c_{2 }= 45, quad c_{3 } = 2240 big (214+7 sqrt{1009},big )big /27, end{aligned}$$</span></div></div><p>and remark that <span>(c_n)</span> is the root of a polynomial of degree <span>(n-1)</span>. We demonstrate that for <span>(n=6,7)</span>, <span>(c_n)</span> are algebraic numbers not expressible as radicals over <span>({{mathbb {Q}}})</span> (and conjecture this is in fact true for general <span>(n ge 6)</span>).</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"26 1","pages":"165 - 201"},"PeriodicalIF":1.4,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00023-024-01451-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Negative Spectrum of Schrödinger Operators with Rapidly Oscillating Potentials","authors":"Larry Read","doi":"10.1007/s00023-024-01457-8","DOIUrl":"10.1007/s00023-024-01457-8","url":null,"abstract":"<div><p>For Schrödinger operators with potentials that are asymptotically homogeneous of degree <span>(-2)</span>, the size of the coupling determines whether it has finite or infinitely many negative eigenvalues. In the latter case, the asymptotic accumulation of these eigenvalues at zero has been determined by Kirsch and Simon. A similar regime occurs for potentials that are not asymptotically monotone but oscillatory. In this case, when the ratio between the amplitude and frequency of oscillation is asymptotically homogeneous of degree <span>(-1)</span>, the coupling determines the finiteness of the negative spectrum. We present a new proof of this fact by making use of a ground-state representation. As a consequence of this approach, we derive an asymptotic formula analogous to that of Kirsch and Simon.</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"26 1","pages":"81 - 97"},"PeriodicalIF":1.4,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00023-024-01457-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Algebraic Localization of Wannier Functions Implies Chern Triviality in Non-periodic Insulators","authors":"Jianfeng Lu,&nbsp;Kevin D. Stubbs","doi":"10.1007/s00023-024-01444-z","DOIUrl":"10.1007/s00023-024-01444-z","url":null,"abstract":"<div><p>For gapped periodic systems (insulators), it has been established that the insulator is topologically trivial (i.e., its Chern number is equal to 0) if and only if its Fermi projector admits an orthogonal basis with finite second moment (i.e., all basis elements satisfy <span>(int |varvec{x}|^2 |w(varvec{x})|^2 ,text {d}{varvec{x}} &lt; infty )</span>). In this paper, we extend one direction of this result to non-periodic gapped systems. In particular, we show that the existence of an orthogonal basis with slightly more decay (<span>(int |varvec{x}|^{2+epsilon } |w(varvec{x})|^2 ,text {d}{varvec{x}} &lt; infty )</span> for any <span>(epsilon &gt; 0)</span>) is a sufficient condition to conclude that the Chern marker, the natural generalization of the Chern number, vanishes.</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"25 8","pages":"3911 - 3926"},"PeriodicalIF":1.4,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Strong Cosmic Censorship for the Spherically Symmetric Einstein–Maxwell-Charged-Klein–Gordon System with Positive (Lambda ): Stability of the Cauchy Horizon and (H^1) Extensions","authors":"Flavio Rossetti","doi":"10.1007/s00023-024-01454-x","DOIUrl":"10.1007/s00023-024-01454-x","url":null,"abstract":"<div><p>We investigate the interior of a dynamical black hole as described by the Einstein–Maxwell-charged-Klein–Gordon system of equations with a cosmological constant, under spherical symmetry. In particular, we consider a characteristic initial value problem where, on the outgoing initial hypersurface, interpreted as the event horizon <span>(mathcal {H}^+)</span> of a dynamical black hole, we prescribe: (a) initial data asymptotically approaching a fixed sub-extremal Reissner–Nordström–de Sitter solution and (b) an exponential Price law upper bound for the charged scalar field. After showing local well-posedness for the corresponding first-order system of partial differential equations, we establish the existence of a Cauchy horizon <span>(mathcal{C}mathcal{H}^+)</span> for the evolved spacetime, extending the bootstrap methods used in the case <span>(Lambda = 0)</span> by Van de Moortel (Commun Math Phys 360:103–168, 2018. https://doi.org/10.1007/s00220-017-3079-3). In this context, we show the existence of <span>(C^0)</span> spacetime extensions beyond <span>(mathcal{C}mathcal{H}^+)</span>. Moreover, if the scalar field decays at a sufficiently fast rate along <span>(mathcal {H}^+)</span>, we show that the renormalized Hawking mass remains bounded for a large set of initial data. With respect to the analogous model concerning an uncharged and massless scalar field, we are able to extend the known range of parameters for which mass inflation is prevented, up to the optimal threshold suggested by the linear analyses by Costa–Franzen (Ann Henri Poincaré 18:3371–3398, 2017. https://doi.org/10.1007/s00023-017-0592-z) and Hintz–Vasy (J Math Phys 58(8):081509, 2017. https://doi.org/10.1063/1.4996575). In this no-mass-inflation scenario, which includes near-extremal solutions, we further prove that the spacetime can be extended across the Cauchy horizon with continuous metric, Christoffel symbols in <span>(L^2_{text {loc}})</span> and scalar field in <span>(H^1_{text {loc}})</span>. By generalizing the work by Costa–Girão–Natário–Silva (Commun Math Phys 361:289–341, 2018. https://doi.org/10.1007/s00220-018-3122-z) to the case of a charged and massive scalar field, our results reveal a potential failure of the Christodoulou–Chruściel version of the strong cosmic censorship under spherical symmetry.</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"26 2","pages":"675 - 753"},"PeriodicalIF":1.4,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00023-024-01454-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}