Semiclassical Setting Research Articles

Discrete heat equation with irregular thermal conductivity and tempered distributional data

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb {Z}^n$ . We establish the well-posedness of such Cauchy problems in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell ^{2}_{s}(\hbar \mathbb {Z}^n)$ , $s \in \mathbb {R}$ , which is enough to prove the well-posedness in the space of tempered distributions $\mathcal {S}'(\hbar \mathbb {Z}^n)$ . Notably, when $s=0$ , we show that for $\hbar \rightarrow 0$ , the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb {R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb {Z}^n$ .

Geometric Invariance of the Semi-classical Calculus on Nilpotent Graded Lie Groups

In this paper, we consider the semi-classical setting constructed on nilpotent graded Lie groups by means of representation theory. Our aim is to analyze the effects of the pull-back by diffeomorphisms on pseudodifferential operators. We restrict to diffeomorphisms that preserve the filtration and prove that they are uniformly Pansu differentiable. We show that the pull-back of a semi-classical pseudodifferential operator by such a diffeomorphism has a semi-classical symbol that is expressed at leading order in terms of the Pansu differential. Finally, we interpret the geometric meaning of this invariance in the setting of filtered manifolds.

Semiclassical analysis and the Agmon-Finsler metric for discrete Schrödinger operators

We study the Agmon estimate and the exponential decay of eigenfunctions for multi-dimensional discrete Schrödinger operators with emphasis on the microlocal analysis on the torus. We first consider the semiclassical setting where semiclassical continuous Schrödinger operators are discretized with the mesh width proportional to the semiclassical parameter. For a general class of potentials, we prove the Agmon estimate for eigenfunctions using the Agmon metric which is a Finsler metric rather than a Riemannian metric. Klein and Rosenberger (2008) proved this by a different argument in the case of a potential minimum. Our argument seems to be simpler. We also prove the Agmon estimate and the optimal anisotropic exponential decay of eigenfunctions for discrete Schrödinger operators in the non-semiclassical standard setting.

Pseudomodes for non-self-adjoint Dirac operators

Depending on the behaviour of the complex-valued electromagnetic potential in the neighbourhood of infinity, pseudomodes of one-dimensional Dirac operators corresponding to large pseudoeigenvalues are constructed. This is a first systematic approach which goes beyond the standard semi-classical setting. Furthermore, this approach results in substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials, including superexponential ones.

Manipulating the dynamics of a Fermi resonance with light. A direct optimal control theory approach

Direct optimal control theory for quantum dynamical problems presents itself as an interesting alternative to the traditional indirect optimal control. The method relies on the first discretize and then optimize paradigm, where a discretization of the dynamical equations leads to a nonlinear optimization problem. It has been applied successfully to the control of a bistable system where the wavepacket had been approximated by a parameterized Gaussian, leading to a semiclassical set of equations of motion (A. R. Ramos Ramos, O. K\"uhn, Front. Phys. 9 (2021) 615168). Motivated by these results, in the present paper we extend the application of the method to the case of exact wavepacket propagation using the example of a generic Fermi-resonance model. In particular we address the question how population of the involved overtone state can be avoided such as to reduce the effect of intramolecular vibrational energy redistribution. A methodological advantage is that direct optimal control theory offers flexibility when choosing the running cost, since there is no need to compute functional derivatives and coupling terms as in the case of indirect optimal control. We exploit this fact to include state populations in the running cost, which allows their optimization.

No Events on Closed Causal Curves

We introduce the Causal Compatibility Conjecture for the Events, Trees, Histories (ETH) approach to Quantum Theory (QT) in the semi-classical setting. We then prove that under the assumptions of the conjecture, points on closed causal curves are physically indistinguishable in the context of the ETH approach to QT and thus the conjecture implies a compatibility of the causal structures even in presence of closed causal curves. As a consequence of this result there is no observation that could be made by an observer to tell any two points on a closed causal curve apart. We thus conclude that closed causal curves have no physical significance in the context of the ETH approach to QT. This is an indication that time travel will not be possible in a full quantum theory of gravity and thus forever remain a fantasy.

Outer entropy equals Bartnik-Bray inner mass and the gravitational ant conjecture

Entropy and energy are found to be closely tied on our quest for quantum gravity. We point out an interesting connection between the recently proposed outer entropy, a coarse-grained entropy defined for a compact spacetime domain motivated by the holographic duality, and the Bartnik-Bray quasilocal mass long known in the mathematics community. In both scenarios, one seeks an optimal spacetime fill-in of a given closed, connected, spacelike, codimension-two boundary. We show that for an outer-minimizing mean-convex surface, the Bartnik-Bray inner mass matches exactly with the irreducible mass corresponding to the outer entropy. The equivalence implies that the area laws derived from the outer entropy are mathematically equivalent as the monotonicity property of the quasilocal mass. It also gives rise to new bounds between entropy and the gravitational energy, which naturally gives the gravitational counterpart to Wall's ant conjecture. We also observe that the equality can be achieved in a conformal flow of metrics, which is structurally similar to the Ceyhan-Faulkner proof of the ant conjecture. We compute the small sphere limit of the outer entropy and it is proportional to the bulk stress tensor as one would expect for a quasilocal mass. Lastly, we discuss some implications of taking quantum matter into consideration in the semiclassical setting.

Concentration of nodal solutions for logarithmic scalar field equations

This paper studies sign-changing solutions and their concentration behaviors of logarithmic scalar field equations in the semiclassical setting. At a local minimum of the potential function we construct an unbounded sequence of sign-changing solutions concentrating near the local minimum. This resembles a localized phenomenon of an unbounded sequence of localized bound states in the studies of nonlinear eigenvalues and eigenfunctions.

Distribution of Resonances in Scattering by Thin Barriers

We study high energy resonances for the operators − Δ ∂ Ω , δ := − Δ + δ ∂ Ω ⊗ V and − Δ ∂ Ω , δ ′ := − Δ + δ ∂ Ω ′ ⊗ V ∂ ν \begin{equation*} -\Delta _{\partial \Omega ,\delta }:=-\Delta +\delta _{\partial \Omega }\otimes V\quad \text {and}\quad -\Delta _{\partial \Omega ,\delta ’}:=-\Delta +\delta _{\partial \Omega }’\otimes V\partial _\nu \end{equation*} where Ω ⊂ R d \Omega \subset \mathbb {R}^d is strictly convex with smooth boundary, V : L 2 ( ∂ Ω ) → L 2 ( ∂ Ω ) V:L^2(\partial \Omega )\to L^2(\partial \Omega ) may depend on frequency, and δ ∂ Ω \delta _{\partial \Omega } is the surface measure on ∂ Ω \partial \Omega . These operators are model Hamiltonians for the quantum corrals studied in Aligia and Lobos (2005), Barr, Zaletel, and Heller (2010), and Crommie et al. (1995) and for leaky quantum graphs in Exner (2008). We give a quantum version of the Sabine Law (Sabine, 1964) from the study of acoustics for both − Δ ∂ Ω , δ -\Delta _{\partial \Omega ,\delta } and − Δ ∂ Ω , δ ′ -\Delta _{\partial \Omega ,\delta ’} . It characterizes the decay rates (imaginary parts of resonances) in terms of the system’s ray dynamics. In particular, the decay rates are controlled by the average reflectivity and chord length of the barrier. For − Δ ∂ Ω , δ -\Delta _{\partial \Omega ,\delta } with Ω \Omega smooth and strictly convex, our results improve those given for general ∂ Ω \partial \Omega in Galkowski and Smith (2015) and are generically optimal. Indeed, we show that for generic domains and potentials there are infinitely many resonances arbitrarily close to the resonance free region found by our theorem. In the case of − Δ ∂ Ω , δ ′ -\Delta _{\partial \Omega ,\delta ’} , the quantum Sabine law gives the existence of a resonance free region that converges to the real axis at a fixed polynomial rate. The size of this resonance free region is optimal in the case of the unit disk in R 2 \mathbb {R}^2 . As far as the author is aware, this is the only class of examples that is known to have resonances converging to the real axis at a fixed polynomial rate but no faster. The proof of our theorem requires several new technical tools. We adapt intersecting Lagrangian distributions from Melrose and Uhlman (1979) to the semiclassical setting and give a description of the kernel of the free resolvent as such a distribution. We also construct a semiclassical version of the Melrose–Taylor parametrix (Melrose and Taylor, unpublished manuscript) for complex energies. We use these constructions to give a complete microlocal description of the single, double, and derivative double layer operators in the case that ∂ Ω \partial \Omega is smooth and strictly convex. These operators are given respectively for x ∈ ∂ Ω x\in \partial \Omega by G ( λ ) f ( x ) a m p ; := ∫ ∂ Ω R 0 ( λ ) ( x , y ) f ( y ) d S ( y ) , N ~ ( λ ) f ( x ) a m p ; := ∫ ∂ Ω ∂ ν y R 0 ( λ ) ( x , y ) f ( y ) d S ( y ) ∂ ν D ℓ ( λ ) f ( x ) a m p ; := ∫ ∂ Ω ∂ ν x ∂ ν y R 0 ( λ ) ( x , y ) f ( y ) d S ( y ) . \begin{align*} G(\lambda )f(x)&:=\int _{\partial \Omega }R_0(\lambda )(x,y)f(y)dS(y)\,,\\ \tilde {N}(\lambda )f(x)&:=\int _{\partial \Omega }\partial _{\nu _y}R_0(\lambda )(x,y)f(y)dS(y)\\ \partial _{\nu }\mathcal {D}\ell (\lambda )f(x)&:=\int _{\partial \Omega }\partial _{\nu _x}\partial _{\nu _y}R_0(\lambda )(x,y)f(y)dS(y)\,. \end{align*} This microlocal description allows us to prove sharp high energy estimates on G G , N ~ \tilde {N} , and ∂ ν D ℓ \partial _{\nu }\mathcal {D}\ell when Ω \Omega is smooth and strictly convex, removing the log losses from the estimates for G G in Galkowski and Smith (2015) and Han and Tacy (2015) and proving a conjecture from Appendix A in Han and Tacy (2015).

Lorentz-violating scalar Hamiltonian and the equivalence principle in a static metric

In this paper, we obtain a nonrelativistic Hamiltonian from the Lorentz-violating (LV) scalar Lagrangian in the minimal standard model extension (SME). The Hamiltonian is obtained by two different methods. One is through the usual ansatz $\mathrm{\ensuremath{\Phi}}(t,\stackrel{\ensuremath{\rightarrow}}{r})={e}^{\ensuremath{-}imt}\mathrm{\ensuremath{\Psi}}(t,\stackrel{\ensuremath{\rightarrow}}{r})$ applied to the LV-corrected Klein-Gordon equation, and the other is the Foldy-Wouthuysen transformation. The consistency of our results is also partially supported by the comparison with the spin-independent part of the fermion Hamiltonian. In this comparison, we can also establish a relation between the set of scalar LV coefficients with their fermion counterparts. Using a pedagogical definition of the weak equivalence principle (WEP), we further point out that the LV Hamiltonian not only necessarily violates universal free fall, which is clearly demonstrated in the geodesic deviation, but also violates WEP in a semiclassical setting. As a bosonic complement, this method can be straightforwardly applicable to the spin-1 case, which shall be useful in the analysis of atomic tests of WEP, such as the case of the ${^{87}\mathrm{Rb}}_{1}$ atom.

The quantization of normal velocity does not concentrate on hypersurfaces

ABSTRACTWe seek to extend work by Christianson–Hassell–Toth [3] on restrictions of Neumann data of Laplacian eigenfunctions to interior hypersurfaces to a general semiclassical setting. In the semiclassical regime the appropriate generalization is to study the restrictions of the function v = ν(x,hD)u where ν(x,hD) is the operator defined by quantizing the normal velocity observable. For the Laplacian where ν is the normal to the hypersurface. We find that provided u is an quasimode of the semiclassical pseudodifferential operator p(x,hD). This statement should be interpreted as a statement of non-concentration for the quantization of normal velocity.

Semiclassical basis sets for the computation of molecular vibrational states.

In this paper, we extend a method recently reported [F. Revuelta et al., Phys. Rev. E 87, 042921 (2013)] for the calculation of the eigenstates of classically highly chaotic systems to cases of mixed dynamics, i.e., those presenting regular and irregular motions at the same energy. The efficiency of the method, which is based on the use of a semiclassical basis set of localized wave functions, is demonstrated by applying it to the determination of the vibrational states of a realistic molecular system, namely, the LiCN molecule.

Unique Continuation for Quasimodes on Surfaces of Revolution: Rotationally invariant Neighbourhoods

We prove a strong conditional unique continuation estimate for irreducible quasimodes in rotationally invariant neighbourhoods on compact surfaces of revolution. The estimate states that Laplace quasimodes which cannot be decomposed as a sum of other quasimodes have L mass bounded below by Cǫλ for any ǫ > 0 on any open rotationally invariant neighbourhood which meets the semiclassical wavefront set of the quasimode. For an analytic manifold, we conclude the same estimate with a lower bound of Cδλ −1+δ for some fixed δ > 0.

Numerical study of fractional nonlinear Schrödinger equations.

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.

Spectral invariants for coupled spin-oscillators

This text deals with inverse spectral theory in a semiclassical setting. Given a quantum system, the haunting question is “What interesting quantities can be discovered on the spectrum that can help to characterize the system ?” The general framework will be semiclassical analysis, and the issue is to recover the classical dynamics from the quantum spectrum. The coupling of a spin and an oscillator is a fundamental example in physics where some nontrivial explicit calculations can be done.

Partial decoherence and thermalization through time-domain ergodicity

An approach, differing from two commonly used methods (the stochastic \SE \ and the master equation \cite {Schlosshauer,BieleA}) but entrenched in the traditional density matrix formalism, is developed in a semi-classical setting, so as to go from the solutions of the time dependent \SE to decohering and thermalized states. This is achieved by utilizing the time-ergodicity, rather than the sampling- (or ensemble-) ergodicity, of physical systems. We introduce the formalism through a study of the Rabi model (a two level system coupled to an oscillator) and show that our semi-classical version exhibits, both qualitatively and quantitatively, many features of state truncation and equilibration \cite {AllahverdyanBN}. We then study the time evolution of two qubits in interaction with a bosonic environment, such that the energy scale of one qubit is much larger, and that of the other much smaller than the environment's energy scale. The small energy qubit decoheres to a mixture, while the high energy qubit is protected through the adiabatic theorem. However, an inter-qubit coupling generates an overall decoherence and leads for some values of the coupling to long term revivals in the state occupations.

Non-Concentration of Quasimodes for Integrable Systems

We consider the possible concentration in phase space of a sequence of eigenfunctions (or, more generally, a quasimode) of an operator whose principal symbol has completely integrable Hamilton flow. The semiclassical wavefront set WF h of such a sequence is invariant under the Hamilton flow. In principle this may allow concentration of WF h along a single closed orbit if all frequencies of the flow are rationally related. We show that, subject to non-degeneracy hypotheses, this concentration may not in fact occur. Indeed, in the two-dimensional case, we show that WF h must fill out an entire Lagrangian torus. The main tools are the spreading of Lagrangian regularity previously shown by the author, and an analysis of higher order transport equations satisfied by the principal symbol of a Lagrangian quasimode. These yield a unique continuation theorem for the principal symbol of Lagrangian quasimode, which is the principal new result of the paper.

Asymptotic analysis of quantum dynamics in crystals: the Bloch-Wigner transform, Bloch dynamics and Berry phase

We study the semi-classical limit of the Schrodinger equation in a crystal in the presence of an external potential and magnetic field. We first introduce the Bloch-Wigner transform and derive the asymptotic equations governing this transform in the semi-classical setting. For the second part, we focus on the appearance of the Berry curvature terms in the asymptotic equations. These terms play a crucial role in many important physical phenomena such as the quantum Hall effect. We give a simple derivation of these terms in different settings using asymptotic analysis.

Gravity from the entropy of light

The holographic principle, considered in a semiclassical setting, is shown to have direct consequences on physics at a fundamental level. In particular, a certain relation is pointed out to be the expression of holography in basic thermodynamics. It is argued moreover that through this relation holography can be recognized to induce gravity, and an expression for the gravitational lensing is obtained in terms of entropy over wavelength of black-body radiation or, at a deeper level, in terms of maximum entropy over the associated space to the elementary bit of information.

Semiclassical second microlocal propagation of regularity and integrable systems

We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifold X with respect to a Lagrangian submanifold of T*X. The construction of the calculus, closely analogous to one performed by Bony in the setting of homogeneous Lagrangians, proceeds via the consideration of a model case, that of the zero section of T*ℝn, and conjugation by appropriate Fourier integral operators. We prove a propagation theorem for the associated wavefront set analogous to Hormander’s theorem for operators of real principal type. As an application, we consider the propagation of Lagrangian regularity on invariant tori for quasimodes (e.g., eigenfunctions) of an operator with completely integrable classical hamiltonian. We prove a secondary propagation result for second wavefront set which implies that even in the (extreme) case of Lagrangian tori with all frequencies rational, provided a nondegeneracy assumption holds, Lagrangian regularity either spreads to fill out a whole torus or holds nowhere locally on it.