Partial Differential Equations, Spectral Theory, and Mathematical Physics. The Ari Laptev Anniversary Volume

Preface
Contents
A non-existence result for a generalized radial Brezis–Nirenbergproblem
Introduction
Non-existence of solutions, via the Pohozaev virial identity, using the conformal Laplacian
Non-existence of solutions using the Rellich–Pohozaev argument and a ``Hardy-type'' inequality
An illustrative example
Bibliography
Friedrichs-type inequalities in arbitrary domains
Introduction
Background and notations
First-order inequalities
Second-order inequalities
Inequalities for the symmetric gradient
Sketches of proofs
Bibliography
Ari Laptev and the Journal of Spectral Theory
Critical magnetic field for 2D magnetic Dirac–Coulomb operators and Hardy inequalities
Introduction and main results
Homogeneous Hardy-like inequalities
The minimization problem
The 2D magnetic Dirac–Coulomb operator with an Aharonov–Bohm magnetic field
Non-relativistic limit
Appendix
Bibliography
The Feshbach–Schur map and perturbation theory
Set-up and result
Perturbation estimates
Application: The ground state energy of the Helium-type ions
The eigenfunctions of the Hydrogen-like Hamiltonian
The numerical approximation of the constants $w_1$, $w_2$, $w_1^{\rm as}$ and $w_2^{\rm as}$
Bibliography
Adiabatic and non-adiabatic evolution of wave packets and applications to initial value representations
Introduction
Propagation of Gaussian states and the Herman–Kluk approximation for scalar equations
Herman–Kluk formula in the adiabatic setting
What about smooth crossings?
A sketchy proof for Herman–Kluk approximations
Bibliography
Length scales for BEC in the dilute Bose gas
Introduction
Energy in small boxes
Localization to small boxes
Facts about the scattering solution
Bibliography
The periodic Lieb–Thirring inequality
The periodic Lieb–Thirring inequality
The one-dimensional integrable case $γ=\frac{3}{2}$
Numerical simulations in 1D and 2D
A minimization problem with entropy
Computation of the density of states of $V_k$
Bibliography
Semiclassical asymptotics for a class of singular Schrödingeroperators
Introduction
Preliminaries
Local asymptotics
From local to global asymptotics
Properties of $P_\nu$
Bibliography
On the spectral properties of the Bloch–Torrey equation in infinite periodically perforated domains
Introduction
The Bloch–Torrey operator in the perforated whole plane
Floquet approach for $y$-periodic problems
The Bloch–Torrey operator in a perforated cylinder continued
Quasi-modes and non-emptiness of the spectrum
Conclusion
Generalized Lax–Milgram theorem
Spectrum and Weyl's sequences
Bibliography
Counting bound states with maximal Fourier multipliers
Introduction
The splitting trick
Bibliography
Sharp dimension estimates of the attractor of the damped2D Euler–Bardina equations
Introduction
Global attractor
Dimension estimate
A sharp lower bound
Appendix
Bibliography
Upper estimates for the electronic density in heavy atoms andmolecules
Introduction
Main intermediate inequality
Estimates of the averaged electronic density
Estimates of the correlation function
Proof of Theorem 1.1
Bibliography
Non-linear Schrödinger equation in a uniform magnetic field
Introduction
Non-linear magnetic Schrödinger equation
The non-linear Pauli equation
Bibliography
Large $|k|$ behavior of d-bar problems for domains with a smoothboundary
Introduction
Main result
Bibliography
Heat kernel estimates for two-dimensional relativistic Hamiltonians with magnetic field
Introduction
Radial magnetic field
Aharonov–Bohm-type magnetic fields
Bibliography
A version of Watson lemma for Laplace integrals and someapplications
Introduction
Proofs of the results
Some corollaries
Bibliography
Wehrl-type coherent state entropy inequalities for $\mathrm{SU}(1,1)$ and its $AX+B$ subgroup
Introduction
Unitary representations and coherent states for the $AX+B$ group
Generalized conjecture for the $AX+B$ group and a partial result
An analytic formulation
Some unitary $\mathrm{SU}(1,1)$ representations and their coherent states
The $\mathrm{SU}(1,1)$ quantum channels
Bibliography
Blow-ups for the Horn–Kapranov parametrization of the classicaldiscriminant
Introduction
Extremal discriminant and factorization of its truncations
Parametrizations of zero sets of discriminants
Blow-ups of Horn–Kapranov parametrizations and proof ofthe theorem
Bibliography
Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case
Introduction
Setting and main results
Some reductions
Geometry considerations
Estimates
Approximation of the weight
The asymptotic formula
Bibliography
Relations between two parts of the spectrum of a Schrödinger operator and other remarks on the absolute continuity of the spectrum ina typical case
Introduction
Proof of Theorem 3.4
Bibliography
Bogoliubov theory for many-body quantum systems
Introduction
Bose gases, energy and dynamics
The correlation energy of a Fermi gas
Dynamics of a Fröhlich polaron
Bibliography
A statistical theory of heavy atoms: Asymptotic behavior of the energy and stability of matter
Introduction
Bounds on the energy
Stability of matter
Bibliography
Homogenization of the higher-order Schrödinger-type equations with periodic coefficients
Introduction
Abstract operator-theoretic scheme
Periodic differential operators in $L_2(\mathbb{R}^d;\mathbb{C}^n)$
Application of the abstract results to $A(\mathbf{k})$
Approximation for the operator exponential of $A$
Homogenization of the Schrödinger-type equation
Bibliography
Trace formulas for the modified Mathieu equation
Introduction
General case
Examples
Existence of the solution $\psi_{1}(x,\lambda)$
Bibliography
Eigenvalue accumulation and bounds for non-selfadjoint matrixdifferential operators related to NLS
Introduction
Auxiliary spectral bounds of independent interest
Invariant subspaces and non-real spectrum
Bibliography
Scattering theory for Laguerre operators
Introduction
Jacobi operators and orthogonal polynomials
Wave operators
Time evolution
Bibliography
List of contributors