@article{13178,
  abstract     = {We consider the large polaron described by the Fröhlich Hamiltonian and study its energy-momentum relation defined as the lowest possible energy as a function of the total momentum. Using a suitable family of trial states, we derive an optimal parabolic upper bound for the energy-momentum relation in the limit of strong coupling. The upper bound consists of a momentum independent term that agrees with the predicted two-term expansion for the ground state energy of the strongly coupled polaron at rest and a term that is quadratic in the momentum with coefficient given by the inverse of twice the classical effective mass introduced by Landau and Pekar.},
  author       = {Mitrouskas, David Johannes and Mysliwy, Krzysztof and Seiringer, Robert},
  issn         = {2050-5094},
  journal      = {Forum of Mathematics},
  pages        = {1--52},
  publisher    = {Cambridge University Press},
  title        = {{Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron}},
  doi          = {10.1017/fms.2023.45},
  volume       = {11},
  year         = {2023},
}

@article{14192,
  abstract     = {For the Fröhlich model of the large polaron, we prove that the ground state energy as a function of the total momentum has a unique global minimum at momentum zero. This implies the non-existence of a ground state of the translation invariant Fröhlich Hamiltonian and thus excludes the possibility of a localization transition at finite coupling.},
  author       = {Lampart, Jonas and Mitrouskas, David Johannes and Mysliwy, Krzysztof},
  issn         = {1572-9656},
  journal      = {Mathematical Physics, Analysis and Geometry},
  keywords     = {Geometry and Topology, Mathematical Physics},
  number       = {3},
  publisher    = {Springer Nature},
  title        = {{On the global minimum of the energy–momentum relation for the polaron}},
  doi          = {10.1007/s11040-023-09460-x},
  volume       = {26},
  year         = {2023},
}

@phdthesis{11473,
  abstract     = {The polaron model is a basic model of quantum field theory describing a single particle
interacting with a bosonic field. It arises in many physical contexts. We are mostly concerned
with models applicable in the context of an impurity atom in a Bose-Einstein condensate as
well as the problem of electrons moving in polar crystals.
The model has a simple structure in which the interaction of the particle with the field is given
by a term linear in the field’s creation and annihilation operators. In this work, we investigate
the properties of this model by providing rigorous estimates on various energies relevant to the
problem. The estimates are obtained, for the most part, by suitable operator techniques which
constitute the principal mathematical substance of the thesis.
The first application of these techniques is to derive the polaron model rigorously from first
principles, i.e., from a full microscopic quantum-mechanical many-body problem involving an
impurity in an otherwise homogeneous system. We accomplish this for the N + 1 Bose gas
in the mean-field regime by showing that a suitable polaron-type Hamiltonian arises at weak
interactions as a low-energy effective theory for this problem.
In the second part, we investigate rigorously the ground state of the model at fixed momentum
and for large values of the coupling constant. Qualitatively, the system is expected to display
a transition from the quasi-particle behavior at small momenta, where the dispersion relation
is parabolic and the particle moves through the medium dragging along a cloud of phonons, to
the radiative behavior at larger momenta where the polaron decelerates and emits free phonons.
At the same time, in the strong coupling regime, the bosonic field is expected to behave purely
classically. Accordingly, the effective mass of the polaron at strong coupling is conjectured to
be asymptotically equal to the one obtained from the semiclassical counterpart of the problem,
first studied by Landau and Pekar in the 1940s. For polaron models with regularized form
factors and phonon dispersion relations of superfluid type, i.e., bounded below by a linear
function of the wavenumbers for all phonon momenta as in the interacting Bose gas, we prove
that for a large window of momenta below the radiation threshold, the energy-momentum
relation at strong coupling is indeed essentially a parabola with semi-latus rectum equal to the
Landau–Pekar effective mass, as expected.
For the Fröhlich polaron describing electrons in polar crystals where the dispersion relation is
of the optical type and the form factor is formally UV–singular due to the nature of the point
charge-dipole interaction, we are able to give the corresponding upper bound. In contrast to
the regular case, this requires the inclusion of the quantum fluctuations of the phonon field,
which makes the problem considerably more difficult.
The results are supplemented by studies on the absolute ground-state energy at strong coupling,
a proof of the divergence of the effective mass with the coupling constant for a wide class of
polaron models, as well as the discussion of the apparent UV singularity of the Fröhlich model
and the application of the techniques used for its removal for the energy estimates.
},
  author       = {Mysliwy, Krzysztof},
  issn         = {2663-337X},
  pages        = {138},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Polarons in Bose gases and polar crystals: Some rigorous energy estimates}},
  doi          = {10.15479/at:ista:11473},
  year         = {2022},
}

@article{10564,
  abstract     = {We study a class of polaron-type Hamiltonians with sufficiently regular form factor in the interaction term. We investigate the strong-coupling limit of the model, and prove suitable bounds on the ground state energy as a function of the total momentum of the system. These bounds agree with the semiclassical approximation to leading order. The latter corresponds here to the situation when the particle undergoes harmonic motion in a potential well whose frequency is determined by the corresponding Pekar functional. We show that for all such models the effective mass diverges in the strong coupling limit, in all spatial dimensions. Moreover, for the case when the phonon dispersion relation grows at least linearly with momentum, the bounds result in an asymptotic formula for the effective mass quotient, a quantity generalizing the usual notion of the effective mass. This asymptotic form agrees with the semiclassical Landau–Pekar formula and can be regarded as the first rigorous confirmation, in a slightly weaker sense than usually considered, of the validity of the semiclassical formula for the effective mass.},
  author       = {Mysliwy, Krzysztof and Seiringer, Robert},
  issn         = {1572-9613},
  journal      = {Journal of Statistical Physics},
  number       = {1},
  publisher    = {Springer Nature},
  title        = {{Polaron models with regular interactions at strong coupling}},
  doi          = {10.1007/s10955-021-02851-w},
  volume       = {186},
  year         = {2022},
}

@article{8705,
  abstract     = {We consider the quantum mechanical many-body problem of a single impurity particle immersed in a weakly interacting Bose gas. The impurity interacts with the bosons via a two-body potential. We study the Hamiltonian of this system in the mean-field limit and rigorously show that, at low energies, the problem is well described by the Fröhlich polaron model.},
  author       = {Mysliwy, Krzysztof and Seiringer, Robert},
  issn         = {1424-0637},
  journal      = {Annales Henri Poincare},
  number       = {12},
  pages        = {4003--4025},
  publisher    = {Springer Nature},
  title        = {{Microscopic derivation of the Fröhlich Hamiltonian for the Bose polaron in the mean-field limit}},
  doi          = {10.1007/s00023-020-00969-3},
  volume       = {21},
  year         = {2020},
}

@article{6840,
  abstract     = {We discuss thermodynamic properties of harmonically trapped
imperfect quantum gases. The spatial inhomogeneity of these systems imposes
a redefinition of the mean-field interparticle potential energy as compared
to the homogeneous case. In our approach, it takes the form a
2N2 ωd, where
N is the number of particles, ω—the harmonic trap frequency, d—system’s
dimensionality, and a is a parameter characterizing the interparticle interaction.
We provide arguments that this model corresponds to the limiting case of
a long-ranged interparticle potential of vanishingly small amplitude. This
conclusion is drawn from a computation similar to the well-known Kac scaling
procedure, which is presented here in a form adapted to the case of an isotropic
harmonic trap. We show that within the model, the imperfect gas of trapped
repulsive bosons undergoes the Bose–Einstein condensation provided d > 1.
The main result of our analysis is that in d = 1 the gas of attractive imperfect
fermions with a = −aF < 0 is thermodynamically equivalent to the gas of
repulsive bosons with a = aB > 0 provided the parameters aF and aB fulfill
the relation aB + aF = . This result supplements similar recent conclusion
about thermodynamic equivalence of two-dimensional (2D) uniform imperfect
repulsive Bose and attractive Fermi gases.},
  author       = {Mysliwy, Krzysztof and Napiórkowski, Marek},
  issn         = {1742-5468},
  journal      = {Journal of Statistical Mechanics: Theory and Experiment},
  number       = {6},
  publisher    = {IOP Publishing},
  title        = {{Thermodynamics of inhomogeneous imperfect quantum gases in harmonic traps}},
  doi          = {10.1088/1742-5468/ab190d},
  volume       = {2019},
  year         = {2019},
}