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Jitse NiesenIn the past few weeks I have been trying to understand the eigenvalue problem (time-independent Schrödinger equation)–𝑢'' + λ (cos 𝑥 + cos τ𝑥) 𝑢 = 𝐸𝑢 where λ is a parameter, 𝐸 is the eigenvalue (blame the physicists for the notation), τ is the golden ratio and the problem is posed on the infinite line. The motivation comes from quasicrystals.Some solutions are localized around a minimum of the potential, but the none of the corresponding eigenvalues are isolated.At higher energies, solutions spread out over the whole line, giving rise to the absolutely continuous spectrum which is a Cantor set.This is wild, at least for me, but partially supported by my own computations and functional analysis results. But I am not fully confident of the former and struggling to understand the latter, so I am not sure whether this picture is complete or even correct. The more I look into it, the less I understand ... any pointers are appreciated.<a href="https://mathstodon.xyz/tags/FunctionalAnalysis" class="mention hashtag" rel="tag">#FunctionalAnalysis</a> <a href="https://mathstodon.xyz/tags/quasicrystal" class="mention hashtag" rel="tag">#quasicrystal</a> <a href="https://mathstodon.xyz/tags/SchrodingerEquation" class="mention hashtag" rel="tag">#SchrodingerEquation</a>

katch wreckin many ways, <a href="https://mastodon.social/tags/mathematical" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#mathematical</a> <a href="https://mastodon.social/tags/physics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#physics</a> is based on the <a href="https://mastodon.social/tags/smoothing" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#smoothing</a> properties of the <a href="https://mastodon.social/tags/integral" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#integral</a> <a href="https://mastodon.social/tags/integrableSystems" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#integrableSystems</a> <a href="https://mastodon.social/tags/mathematicalPhysics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#mathematicalPhysics</a> <a href="https://mastodon.social/tags/complexAnalysis" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#complexAnalysis</a> <a href="https://mastodon.social/tags/spectralTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#spectralTheory</a> <a href="https://mastodon.social/tags/harmonicAnalysis" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#harmonicAnalysis</a> <a href="https://mastodon.social/tags/functionalAnalysis" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#functionalAnalysis</a> <a href="https://mastodon.social/tags/mathematicalAnalysis" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#mathematicalAnalysis</a> <a href="https://mastodon.social/tags/differentialEquations" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#differentialEquations</a> <a href="https://mastodon.social/tags/ODEs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ODEs</a> <a href="https://mastodon.social/tags/PDEs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PDEs</a> <a href="https://mastodon.social/tags/SDEs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#SDEs</a> <a href="https://mastodon.social/tags/DEs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DEs</a> <a href="https://mastodon.social/tags/equations" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#equations</a> <a href="https://mastodon.social/tags/BrownianMotion" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#BrownianMotion</a> <a href="https://mastodon.social/tags/LangevinDynamics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#LangevinDynamics</a> <a href="https://mastodon.social/tags/dynamics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#dynamics</a> <a href="https://mastodon.social/tags/Langevin" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Langevin</a> <a href="https://mastodon.social/tags/StochasticDifferentialEquations" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#StochasticDifferentialEquations</a> <a href="https://mastodon.social/tags/StochasticProcesses" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#StochasticProcesses</a> <a href="https://mastodon.social/tags/WienerProcess" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#WienerProcess</a> <a href="https://mastodon.social/tags/OrnsteinUhlenbeck" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#OrnsteinUhlenbeck</a> <a href="https://mastodon.social/tags/HarmonicOscillator" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#HarmonicOscillator</a> <a href="https://mastodon.social/tags/WaveEquation" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#WaveEquation</a> <a href="https://mastodon.social/tags/Newton" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Newton</a> <a href="https://mastodon.social/tags/Newtonian" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Newtonian</a> <a href="https://mastodon.social/tags/Maxwell" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Maxwell</a> <a href="https://mastodon.social/tags/Einstein" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Einstein</a>

katch wreck`To prove <a href="https://mastodon.social/tags/boundedness" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#boundedness</a> on Lp spaces, Calderón and Zygmund introduced a method of decomposing L1 functions, generalising the rising sun lemma of F. Riesz. This method showed that the operator defined a continuous <a href="https://mastodon.social/tags/operator" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#operator</a> from L1 to the space of <a href="https://mastodon.social/tags/functions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#functions</a> of weak L1. The Marcinkiewicz interpolation theorem and duality then implies that the <a href="https://mastodon.social/tags/singular" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#singular</a> <a href="https://mastodon.social/tags/integral" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#integral</a> operator is bounded on all Lp for 1 <a href="https://en.wikipedia.org/wiki/Singular_integral_operators_of_convolution_type#General_theory" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://en.wikipedia.org/wiki/Singular_integral_operators_of_convolution_type#General_theory</a><a href="https://mastodon.social/tags/functionalAnalysis" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#functionalAnalysis</a> <a href="https://mastodon.social/tags/mathematics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#mathematics</a> <a href="https://mastodon.social/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a>

Atexjamyoutube.com/<a href="https://calckey.social/@xenaproject" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@xenaproject</a> for <a href="https://calckey.social/tags/mathematics" rel="nofollow noopener noreferrer" target="_blank">#mathematics</a> <a href="https://calckey.social/tags/mathsbasics" rel="nofollow noopener noreferrer" target="_blank">#mathsbasics</a>#mathematicallogic <a href="https://calckey.social/tags/sets" rel="nofollow noopener noreferrer" target="_blank">#sets</a> <a href="https://calckey.social/tags/logic" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://calckey.social/tags/topology" rel="nofollow noopener noreferrer" target="_blank">#topology</a> <a href="https://calckey.social/tags/numbertheory" rel="nofollow noopener noreferrer" target="_blank">#numbertheory</a> <a href="https://calckey.social/tags/linearalgebra" rel="nofollow noopener noreferrer" target="_blank">#linearalgebra</a> <a href="https://calckey.social/tags/probability" rel="nofollow noopener noreferrer" target="_blank">#probability</a> <a href="https://calckey.social/tags/measuretheory" rel="nofollow noopener noreferrer" target="_blank">#measuretheory</a> <a href="https://calckey.social/tags/functionalanalysis" rel="nofollow noopener noreferrer" target="_blank">#functionalanalysis</a>

Erwin Schrödinger InstituteAre you keen to read about results in <a href="https://mathstodon.xyz/tags/MathematicalPhysics" class="mention hashtag" rel="tag">#MathematicalPhysics</a> and Analysis of <a href="https://mathstodon.xyz/tags/PDEs" class="mention hashtag" rel="tag">#PDEs</a>? Take a look at the paper of our previous workshop participant! 🧐<a href="https://mathstodon.xyz/tags/FunctionalAnalysis" class="mention hashtag" rel="tag">#FunctionalAnalysis</a> <a href="https://mathstodon.xyz/tags/SpectralTheory" class="mention hashtag" rel="tag">#SpectralTheory</a> <a href="https://fediscience.org/@univienna" class="u-url mention">@univienna</a> <a href="https://arxiv.org/pdf/2303.04527.pdf" target="_blank" rel="nofollow noopener noreferrer" translate="no">https://arxiv.org/pdf/2303.04527.pdf</a>

Erwin Schrödinger InstituteWe thank Prof Elliott Lieb for taking a long journey from <a href="https://hci.social/@princeton" class="u-url mention">@princeton</a> Princeton University and giving us an insight into his research work. 🔎 He is a living example of that the age is only a number and you can still accomplish a lot at the age of 91. 🎉 Watch his lecture on YouTube 🔜 www.youtube.com/@ESIVienna<a href="https://mathstodon.xyz/tags/MathematicalPhysics" class="mention hashtag" rel="tag">#MathematicalPhysics</a> <a href="https://mathstodon.xyz/tags/Bosegas" class="mention hashtag" rel="tag">#Bosegas</a> <a href="https://mathstodon.xyz/tags/condensedmatter" class="mention hashtag" rel="tag">#condensedmatter</a> <a href="https://mathstodon.xyz/tags/statisticalmechanics" class="mention hashtag" rel="tag">#statisticalmechanics</a> <a href="https://mathstodon.xyz/tags/statmech" class="mention hashtag" rel="tag">#statmech</a> <a href="https://mathstodon.xyz/tags/functionalanalysis" class="mention hashtag" rel="tag">#functionalanalysis</a><a href="https://fediscience.org/@univienna" class="u-url mention">@univienna</a>

Jane AdamsToday in diagrams I liked: This "unit circle" imagination of Lp space for different values of p. Twas a meme that brought me down this Wikipedia rabbit hole but I have been staring at this picture for a very long while now 😂 <a href="https://en.m.wikipedia.org/wiki/Lp_space" rel="nofollow noopener noreferrer" target="_blank">https://en.m.wikipedia.org/wiki/Lp_space</a>An example is given of where Euclidean distance falls short: taxi drivers need to use rectilinear distance in gridded cities! <a href="https://vis.social/tags/illustration" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#illustration</a> <a href="https://vis.social/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a> <a href="https://vis.social/tags/computerScience" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#computerScience</a> <a href="https://vis.social/tags/topology" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#topology</a> <a href="https://vis.social/tags/distance" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#distance</a> <a href="https://vis.social/tags/functionalAnalysis" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#functionalAnalysis</a> <a href="https://vis.social/tags/learning" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#learning</a> <a href="https://vis.social/tags/mathematics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#mathematics</a> <a href="https://vis.social/tags/diagram" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#diagram</a> <a href="https://vis.social/tags/design" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#design</a>

maralornI can read C. I have never done any C#. I can write passable C++. But my expertise is in C*.<a href="https://chaos.social/tags/functionalAnalysis" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#functionalAnalysis</a> <a href="https://chaos.social/tags/mathematicalPhysics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#mathematicalPhysics</a>

paureaAnyone can give me references to study seminorms with unit balls which are convex sets? In particular, the relationship with seminorms with unit balls which are Minkowski sums of said convex sets and their symmetries. What properties can be derived from the original seminorms and so on... Books, papers, anything? <a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#math</a> <a href="https://mathstodon.xyz/tags/norm" class="mention hashtag" rel="tag">#norm</a> <a href="https://mathstodon.xyz/tags/seminorm" class="mention hashtag" rel="tag">#seminorm</a> <a href="https://mathstodon.xyz/tags/minkowski" class="mention hashtag" rel="tag">#minkowski</a> <a href="https://mathstodon.xyz/tags/FunctionalAnalysis" class="mention hashtag" rel="tag">#FunctionalAnalysis</a>

MeredithAnswering a good question about my <a href="https://mathstodon.xyz/tags/DirichletSeries" class="mention hashtag" rel="tag">#DirichletSeries</a> <a href="https://mathstodon.xyz/tags/FunctionalAnalysis" class="mention hashtag" rel="tag">#FunctionalAnalysis</a> thread from the other day, I actually linked to some references if folks are curious <a href="https://mathstodon.xyz/@meresar_math/109309318624234932" target="_blank" rel="nofollow noopener noreferrer" translate="no">https://mathstodon.xyz/@meresar_math/109309318624234932</a>

MeredithI have some <a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#math</a> thoughts to add to this <a href="https://mathstodon.xyz/tags/twitter" class="mention hashtag" rel="tag">#twitter</a> thread about dense flows on the torus<a href="https://twitter.com/math_vet/status/1589680817101365248" target="_blank" rel="nofollow noopener noreferrer" translate="no">https://twitter.com/math_vet/status/1589680817101365248</a>You can do this in more dimensions (and in infinitely many, you can connect things to Dirichlet series)!<a href="https://mathstodon.xyz/tags/DirichletSeries" class="mention hashtag" rel="tag">#DirichletSeries</a> <a href="https://mathstodon.xyz/tags/FunctionalAnalysis" class="mention hashtag" rel="tag">#FunctionalAnalysis</a> <a href="https://mathstodon.xyz/tags/FunctionTheory" class="mention hashtag" rel="tag">#FunctionTheory</a>

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