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SUMMARY:Eric Seré
DTSTART;VALUE=DATE-TIME:20210614T133000Z
DTEND;VALUE=DATE-TIME:20210614T143000Z
DTSTAMP;VALUE=DATE-TIME:20210612T234934Z
UID:MCQM21/1
DESCRIPTION:Title: D
irac-Coulomb operators with general charge distribution: results and open
problems\nby Eric Seré as part of Mathematical Challenges in Quantum
Mechanics 2021 Workshop\n\n\nAbstract\nThis talk is based on joint works w
ith M.J. Esteban and M. Lewin. Consider an electron moving in the attracti
ve Coulomb potential generated by a non-negative finite measure representi
ng an external charge density. If the total charge is fixed\, it is well k
nown that the lowest eigenvalue of the corresponding Schrodinger operator
is minimized when the measure is a delta. We investigate the conjecture th
at the same holds for the relativistic Dirac-Coulomb operator. First we gi
ve conditions ensuring that this operator has a natural self-adjoint reali
sation and that its eigenvalues are given by min-max formulas. Then we def
ine a critical charge such that\, if the total charge is fixed below it\,
then there exists a measure minimising the first eigenvalue of the Dirac-C
oulomb operator. Moreover this optimal measure concentrates on a compact s
et of Lebesgue measure zero. The last property is proved using a new uniqu
e continuation principle for Dirac operators.\n
LOCATION:https://researchseminars.org/talk/MCQM21/1/
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BEGIN:VEVENT
SUMMARY:Michael Weinstein
DTSTART;VALUE=DATE-TIME:20210614T150000Z
DTEND;VALUE=DATE-TIME:20210614T160000Z
DTSTAMP;VALUE=DATE-TIME:20210612T234934Z
UID:MCQM21/2
DESCRIPTION:Title: T
ight binding approximation of continuum 2D quantum materials\nby Micha
el Weinstein as part of Mathematical Challenges in Quantum Mechanics 2021
Workshop\n\n\nAbstract\nWe consider 2D quantum materials\, modeled by a co
ntinuum Schroedinger operator whose potential\nis composed of an array of
identical potential wells centered on the vertices of a discrete subset\,
\\Omega\, of the plane. \nWe study the low-lying spectrum in the regime of
very deep potential wells.\n\nWe present results on scaled resolvent norm
convergence to a discrete (tight-binding) operator and\, \nin the transla
tion invariant case\, corresponding results on the scaled convergence of l
ow-lying dispersion surfaces.\nExamples include the single electron model
for bulk graphene ($\\Omega$=honeycomb lattice)\, and \na sharply terminat
ed graphene half-space\, interfaced with the vacuum along an arbitrary lin
e-cut. \nWe also apply our methods to the case of strong constant perpendi
cular magnetic fields. \nThis is joint work with CL Fefferman and J Shapir
o.\n\nA detailed analysis of the spectrum of the limiting tight binding mo
del on a honeycomb lattice\, which is terminated along an arbitrary ration
al line-cut (joint work with CL Fefferman and S Fliss)\, will be presented
in the upcoming lecture of CL Fefferman.\n
LOCATION:https://researchseminars.org/talk/MCQM21/2/
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BEGIN:VEVENT
SUMMARY:Ari Laptev
DTSTART;VALUE=DATE-TIME:20210615T133000Z
DTEND;VALUE=DATE-TIME:20210615T143000Z
DTSTAMP;VALUE=DATE-TIME:20210612T234934Z
UID:MCQM21/3
DESCRIPTION:Title: S
ymmetry Results in Two-Dimensional Inequalities for Aharonov–Bohm Magnet
ic Fields\nby Ari Laptev as part of Mathematical Challenges in Quantum
Mechanics 2021 Workshop\n\n\nAbstract\nWe study functional and spectral p
roperties of perturbations of a magnetic second order differential operato
r on a circle.\n\nThis operator appears when considering the restriction t
o the unit circle of a two dimensional Schrödinger operator with the Bohm
-Aharonov vector potential.\n\nWe prove some Hardy-type inequalities and s
harp Keller-Lieb-Thirring inequalities.\n
LOCATION:https://researchseminars.org/talk/MCQM21/3/
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BEGIN:VEVENT
SUMMARY:Charles Fefferman
DTSTART;VALUE=DATE-TIME:20210615T150000Z
DTEND;VALUE=DATE-TIME:20210615T160000Z
DTSTAMP;VALUE=DATE-TIME:20210612T234934Z
UID:MCQM21/4
DESCRIPTION:Title: G
raphene edge states in a tight binding model\nby Charles Fefferman as
part of Mathematical Challenges in Quantum Mechanics 2021 Workshop\n\n\nAb
stract\nWe study a standard tight binding model of graphene\, sharply\nter
minated along an edge. It is well known that zero energy (“flat band”)
\n edge states arise for a "zigzag" edge\, while an "armchair" edge\ngives
rise to no edge states.\n\nWe present joint work with S. Fliss and M. Wei
nstein that determines\nwhich rational edges give rise to flat band edge s
tates\, and exhibits\nformulas for such edge states when they exist. The j
oint work includes\nalso preliminary results on non-flat-band edge states.
\n\nThanks to results presented in Michael Weinstein's lecture\, flat band
s\nfor a tight binding model give rise to almost flat band edge states\nfo
r a continuum model.\n
LOCATION:https://researchseminars.org/talk/MCQM21/4/
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