Two qubits is the simplest system where the notions of separable and entangled states and entangl... more Two qubits is the simplest system where the notions of separable and entangled states and entanglement witnesses first appear. We give a three dimensional geometric description of these notions. This description however carries no quantitative information on the measure of entanglement. A four dimensional description captures also the entanglement measure. We give a neat formula for the Bell states which leads to a slick proof of the fundamental teleportation identity. We describe optimal distillation of two qubits geometrically and present a simple geometric proof of the Peres-Horodecki separability criterion.
We study the forces that act on a point flux carrying an integral number of flux units in quantum... more We study the forces that act on a point flux carrying an integral number of flux units in quantum Hall fluids. Forces due to external fields, Lorentz and Magnus type forces, and the forces due to mutual interaction of point fluxes are considered. The forces are related to the adiabatic curvature associated with families of Landau Hamiltonians. The problem displays distinct features of the quantum Hall fluids with point fluxes on the plane and on the torus, which, however, agree at the "thermodynamic" limit.
We derive the relativistically exact Eikonal equation for ring interferometers undergoing deforma... more We derive the relativistically exact Eikonal equation for ring interferometers undergoing deformations. For ring interferometers that undergo slow deformation we describe the two leading terms in the adiabatic expansion of the phase shift. The leading term is independent of the refraction index n and is given by a line integral generalizing results going back to Sagnac [21, 28, 18] for non-deforming interferometers to all orders in β = |v|/c. In the non-relativistic limit this term is O(β). The next term in the adiabaticity has the form of a double integral, it is of order β 0 and depends on the refractive index n. It accounts for non-reciprocity due to changing circumstances in the fiber. The adiabatic correction is often comparable to the Sagnac term. In particular, this is the case in Fizeau's interferometer. Besides providing a mathematical framework that puts all ring interferometers under a single umbrella, our results generalize and strengthen results of [18, 24] to fibers with chromatic dispersion.
Distributed quantum computing with classical communications allows to relieve some of the limitat... more Distributed quantum computing with classical communications allows to relieve some of the limitations on the number of qubits and mitigate the noise in quantum computers. We give an algorithm that transforms a quantum circuit on a single processor to equivalent circuits on distributed processors. We address the quantum advantage of distributed circuits for the Grover search, Simon's and the Deutsch-Jozsa problems. In the case of Grover the quantum advantage of distributed computing remains the same, i.e. O(√(N)). In the case of Simon it remains exponential, but the complexity deteriorates from O(n) to O(n^2), where n = log_2(N). The distributed Deutsch-Jozsa deteriorates to being probabilistic but retains a quantum advantage over classical random sampling: A single quantum query gives the same error as O(n) random sampling. In section 5 we describe an experiment with the IBMQ5 machines that illustrates the advantages of distributed Grover search.
The Hofstadter butterfly is viewed as a quantum phase diagram with infinitely many phases, labele... more The Hofstadter butterfly is viewed as a quantum phase diagram with infinitely many phases, labeled by their (integer) Hall conductance, and a fractal structure. We describe various properties of this phase diagram: We establish Gibbs phase rules; count the number of components of each phase, and characterize the set of multiple phase coexistence.
After half a decade since the appearance of our article [1] we sill regularly receive questions a... more After half a decade since the appearance of our article [1] we sill regularly receive questions and remarks about some formulas and expressions in text. Furthermore Dr. M. Hϋbner (Leipzig) points out that some term in Eqs. (2.12), (2.14), (2.16) vanishes. So we write this erratum: 1. On p. 37 in the formula just after Eq. (1.3) there is a minus sign missing:
Using operator algebraic methods we show that the moment generating function of charge transport ... more Using operator algebraic methods we show that the moment generating function of charge transport in a system with infinitely many non-interacting Fermions is given by a determinant of a certain operator in the one-particle Hilbert space. The formula is equivalent to a formula of Levitov and Lesovik in the finite dimensional case and may be viewed as its regularized form in general. Our result embodies two tenets often realized in mesoscopic physics, namely, that the transport properties are essentially independent of the length of the leads and of the depth of the Fermi sea.
Electron-electron interactions have strong effects on the low-energy excitations of a one-dimensi... more Electron-electron interactions have strong effects on the low-energy excitations of a one-dimensional metal. Luttinger liquid theory, which is supposed to describe this situation, predicts, among other things, that an injected electron will split into separate charge and spin excitations, which propagate at different velocities. We shall review some experiments and theoretical analyses where spin-charge separation can have manifest consequences, including discussion of the spin-incoherent regime, which can occur in low-density ...
We study the relative index of two orthogonal infinite dimensional projections which, in the fini... more We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.
This paper completes the proof of the following statement, known as the Ten Martini Problem. The ... more This paper completes the proof of the following statement, known as the Ten Martini Problem. The spectrum of the almost Mathieu operator (Hψ)(n) = ψ(n + 1) + ψ(n − 1) + 2λ cos(2ψ(θ + nα))ψ(n) in ℓ 2 (Z) is nowhere dense, provided that λ = 0 and α ∈ Q (in these exceptional cases, the potential is periodic and the spectrum of the operator is known not to be nowhere dense). This result was known due to earlier works for a large set of parameter values: periodic approximation works for Liouville α [J. Béllissard and B. Simon, J. Funct. Anal. 48 (1982), no. 3, 408–419; MR0678179 (84h:81019); M. D. Choi, G. A. Elliott and N. Yui, Invent. Math. 99
This paper is about adiabatic transport in quantum pumps. The notion of "energy shift", a self-ad... more This paper is about adiabatic transport in quantum pumps. The notion of "energy shift", a self-adjoint operator dual to the Wigner time delay, plays a role in our approach: It determines the current, the dissipation, the noise and the entropy currents in quantum pumps. We discuss the geometric and topological content of adiabatic transport and show that the mechanism of Thouless and Niu for quantized transport via Chern numbers cannot be realized in quantum pumps where Chern numbers necessarily vanish.
Adiabatic evolutions with a gap condition have, under a range of circumstances, exponentially sma... more Adiabatic evolutions with a gap condition have, under a range of circumstances, exponentially small tails that describe the leaking out of the spectral subspace. Adiabatic evolutions without a gap condition do not seem to have this feature in general. This is a known fact for eigenvalue crossing. We show that this is also the case for eigenvalues at the threshold of the continuous spectrum by considering the Friedrichs model.
I explain the thermodynamic significance, the duality and open problems associated with the two c... more I explain the thermodynamic significance, the duality and open problems associated with the two colored butterflies shown in figures 1 and 4.
When basic tools of quantum information are applied to the quantum tomography data presented in [... more When basic tools of quantum information are applied to the quantum tomography data presented in [1], none of their devices appears to be a source of entangled photons. In a paper titled “A semiconductor quantum source of triggered entangled photon pairs ” Stevenson et. al. [1] claim to find evidence for the emission of polarization entangled photons from certain quantum dots. A density matrix completely specifies all properties of the quantum state [2]. Using quantum tomography [3], the authors of [1] construct the density matrices representing the polarization state of the photons emitted by each of their devices. We show that subjecting their quantum states to the basic definition of entanglement leads to the inevitable conclusion that none of their devices produced entangled light. In all the dots investigated in [1], quantum tomography yielded real density matrices of the form ρ = α 0 0 γ
We provide a counterexample to the universal paramagnetism conjecture of Hogreve, Schrader and Se... more We provide a counterexample to the universal paramagnetism conjecture of Hogreve, Schrader and Seiler. The counterexample is based on the Bohm—Aharonov effect. Several years ago, one of us [11proved an inequal- (eq. (18) of ref. [41)that ity expressingthe universal diamagnetic tendency of spinless bosons. For a single particle in external (local) Tr(exp(—j3H 2(a, V))) ~ ‘ Tr(exp(—~3H2(a = 0, V))) (6) electric potential V and magnetic potential a, the in- and, in particular, that equality can be expressed
We announce three new rigorous results for the quantum mechanical hydrogen atom in constant magne... more We announce three new rigorous results for the quantum mechanical hydrogen atom in constant magnetic field: (i) Borel summability of the small field perturbation series, (ii) detailed large field asymptotics, and (iii) non-degeneracy of the ground state ti~and a proof that it hasL~t~o = 0 for all values of the field. The weak field Zeeman effect [1] in simple atoms A =-4(r X B); B = (0,0, B) (2) was one of the earliest problems studied [2] in quantum mechanics. More recently, Ruderman [3] and Theorem 1. Let E~(O)be any negative eigenvalue of then others [4] discussed the analogous problem in the HydrogenHamiltonian H(0). Then there is an eigensuper-strong magnetic fields of the type encountered value [10] En(B) of H(B) for B small which is the Borel in neutron stars. It is perhaps surprising that any prob- sum [11,12] of the Rayleigh-Schrödinger perturbation lems remain open for such a well studied theory but coefficients for En [10]. there are some unresolved theoretical questions ...
Abstract. We study the Lyaponov exponent yA(E) of (hu)(n) = u(n + 1) + u(n- l)+AV(n)u(n) in the l... more Abstract. We study the Lyaponov exponent yA(E) of (hu)(n) = u(n + 1) + u(n- l)+AV(n)u(n) in the limit as A +a where V is a suitable random potential. We prove that yA(E)-ln A as A +CO uniformly as E/A runs through compact sets. We also describe a formal expansion (to order A-*) for random and almost periodic potentials. In this note, we study one-dimensional tight binding Hamiltonians h = ho + A V where (hou)(n) = u(n + 1) +u(n- 1). We are interested in the cases where V is either random or almost periodic. By random, we mean that V(n) is a family of identically distributed independent random variables with density P(y)dy where P is bounded with bounded support. In the random case, we will succeed in identifying the first few terms in the large A behaviour of the Lyaponov exponent. For the almost periodic case only a formal large A expansion is obtained. Explicitly, we let yA (E) be the Lyaponov exponent for h, i.e. where 1 yA(E) = lim-lnllM,,(u)...Ml(w)ll n+m n (1) The limit exists...
We study the relation of the adiabatic curvature associated to scattering states and the scatteri... more We study the relation of the adiabatic curvature associated to scattering states and the scattering matrix. We show that there cannot be any formula relating the two locally. However, the rst Chern number, which is proportional to the integral of the curvature, can be computed by integrating a 3-form constructed from the Smatrix. Similar formulas relate higher Chern classes to integrals of higher degree forms constructed from scattering data. We show that level crossings of the on-shell S-matrix can be assigned an index so that the rst Chern number of the scattering states is the sum of the indices. We construct an example which is the natural scattering analog of Berry's spin 1/2 Hamiltonian.
Incommensurate perturbations of classical orbits lead to an almost periodic Hill's operator ... more Incommensurate perturbations of classical orbits lead to an almost periodic Hill's operator whose spectrum, it is argued, is a Cantor set, but one with large Lebesgue measure. Applied to the rings of Saturn, this implies that the complex groove structure in the rings approximates a Cantor set. The possible relevance of the sun in producing 'side gaps' which magnify the apparent gap size is also emphasized
I study the width of the Wannier ladder states, i.e, Bloch electrons in external homogeneous fiel... more I study the width of the Wannier ladder states, i.e, Bloch electrons in external homogeneous field. For periodic potentials with a finite number of gaps, a formula for the width is obtained showing that the width vanishes exponentially fast with the field in accordance with Zener tunneling. The case of infinitely many gaps is studied qualitatively, and it is argued that although the width decreases exponentially "on the average," the detailed bahavior is very complicated. In particular the width oscillates over different orders of magnitude as the field changes slightly. The oscillations are a consequence of a resonance phenomenon. 1. THE PROBLEM * Work supported in part by USNSF MCS-78-01885. I Another possible interpretation of the Hamiltonian is for neutrons in crystals under a gravitational field. A discussion of gravitational interference effects for neutrons is given in Ref. 1281 and references therein. 2 The units are 2m = h = a/2x = 1, a the lattice spacing. The unit of charge is e* = nh*/ma and the unit of force xh*/ma3. ' It is instructive to contrast (I. I) with H = p + V(x)-fx. H is unitarily equivalent top (see footnote (5)) and has no resonances although it has the same symmetry properties as (I. I). The catch is that p + V(x) has no gaps in the spectrum and does not Bragg scatter.
Two qubits is the simplest system where the notions of separable and entangled states and entangl... more Two qubits is the simplest system where the notions of separable and entangled states and entanglement witnesses first appear. We give a three dimensional geometric description of these notions. This description however carries no quantitative information on the measure of entanglement. A four dimensional description captures also the entanglement measure. We give a neat formula for the Bell states which leads to a slick proof of the fundamental teleportation identity. We describe optimal distillation of two qubits geometrically and present a simple geometric proof of the Peres-Horodecki separability criterion.
We study the forces that act on a point flux carrying an integral number of flux units in quantum... more We study the forces that act on a point flux carrying an integral number of flux units in quantum Hall fluids. Forces due to external fields, Lorentz and Magnus type forces, and the forces due to mutual interaction of point fluxes are considered. The forces are related to the adiabatic curvature associated with families of Landau Hamiltonians. The problem displays distinct features of the quantum Hall fluids with point fluxes on the plane and on the torus, which, however, agree at the "thermodynamic" limit.
We derive the relativistically exact Eikonal equation for ring interferometers undergoing deforma... more We derive the relativistically exact Eikonal equation for ring interferometers undergoing deformations. For ring interferometers that undergo slow deformation we describe the two leading terms in the adiabatic expansion of the phase shift. The leading term is independent of the refraction index n and is given by a line integral generalizing results going back to Sagnac [21, 28, 18] for non-deforming interferometers to all orders in β = |v|/c. In the non-relativistic limit this term is O(β). The next term in the adiabaticity has the form of a double integral, it is of order β 0 and depends on the refractive index n. It accounts for non-reciprocity due to changing circumstances in the fiber. The adiabatic correction is often comparable to the Sagnac term. In particular, this is the case in Fizeau's interferometer. Besides providing a mathematical framework that puts all ring interferometers under a single umbrella, our results generalize and strengthen results of [18, 24] to fibers with chromatic dispersion.
Distributed quantum computing with classical communications allows to relieve some of the limitat... more Distributed quantum computing with classical communications allows to relieve some of the limitations on the number of qubits and mitigate the noise in quantum computers. We give an algorithm that transforms a quantum circuit on a single processor to equivalent circuits on distributed processors. We address the quantum advantage of distributed circuits for the Grover search, Simon's and the Deutsch-Jozsa problems. In the case of Grover the quantum advantage of distributed computing remains the same, i.e. O(√(N)). In the case of Simon it remains exponential, but the complexity deteriorates from O(n) to O(n^2), where n = log_2(N). The distributed Deutsch-Jozsa deteriorates to being probabilistic but retains a quantum advantage over classical random sampling: A single quantum query gives the same error as O(n) random sampling. In section 5 we describe an experiment with the IBMQ5 machines that illustrates the advantages of distributed Grover search.
The Hofstadter butterfly is viewed as a quantum phase diagram with infinitely many phases, labele... more The Hofstadter butterfly is viewed as a quantum phase diagram with infinitely many phases, labeled by their (integer) Hall conductance, and a fractal structure. We describe various properties of this phase diagram: We establish Gibbs phase rules; count the number of components of each phase, and characterize the set of multiple phase coexistence.
After half a decade since the appearance of our article [1] we sill regularly receive questions a... more After half a decade since the appearance of our article [1] we sill regularly receive questions and remarks about some formulas and expressions in text. Furthermore Dr. M. Hϋbner (Leipzig) points out that some term in Eqs. (2.12), (2.14), (2.16) vanishes. So we write this erratum: 1. On p. 37 in the formula just after Eq. (1.3) there is a minus sign missing:
Using operator algebraic methods we show that the moment generating function of charge transport ... more Using operator algebraic methods we show that the moment generating function of charge transport in a system with infinitely many non-interacting Fermions is given by a determinant of a certain operator in the one-particle Hilbert space. The formula is equivalent to a formula of Levitov and Lesovik in the finite dimensional case and may be viewed as its regularized form in general. Our result embodies two tenets often realized in mesoscopic physics, namely, that the transport properties are essentially independent of the length of the leads and of the depth of the Fermi sea.
Electron-electron interactions have strong effects on the low-energy excitations of a one-dimensi... more Electron-electron interactions have strong effects on the low-energy excitations of a one-dimensional metal. Luttinger liquid theory, which is supposed to describe this situation, predicts, among other things, that an injected electron will split into separate charge and spin excitations, which propagate at different velocities. We shall review some experiments and theoretical analyses where spin-charge separation can have manifest consequences, including discussion of the spin-incoherent regime, which can occur in low-density ...
We study the relative index of two orthogonal infinite dimensional projections which, in the fini... more We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.
This paper completes the proof of the following statement, known as the Ten Martini Problem. The ... more This paper completes the proof of the following statement, known as the Ten Martini Problem. The spectrum of the almost Mathieu operator (Hψ)(n) = ψ(n + 1) + ψ(n − 1) + 2λ cos(2ψ(θ + nα))ψ(n) in ℓ 2 (Z) is nowhere dense, provided that λ = 0 and α ∈ Q (in these exceptional cases, the potential is periodic and the spectrum of the operator is known not to be nowhere dense). This result was known due to earlier works for a large set of parameter values: periodic approximation works for Liouville α [J. Béllissard and B. Simon, J. Funct. Anal. 48 (1982), no. 3, 408–419; MR0678179 (84h:81019); M. D. Choi, G. A. Elliott and N. Yui, Invent. Math. 99
This paper is about adiabatic transport in quantum pumps. The notion of "energy shift", a self-ad... more This paper is about adiabatic transport in quantum pumps. The notion of "energy shift", a self-adjoint operator dual to the Wigner time delay, plays a role in our approach: It determines the current, the dissipation, the noise and the entropy currents in quantum pumps. We discuss the geometric and topological content of adiabatic transport and show that the mechanism of Thouless and Niu for quantized transport via Chern numbers cannot be realized in quantum pumps where Chern numbers necessarily vanish.
Adiabatic evolutions with a gap condition have, under a range of circumstances, exponentially sma... more Adiabatic evolutions with a gap condition have, under a range of circumstances, exponentially small tails that describe the leaking out of the spectral subspace. Adiabatic evolutions without a gap condition do not seem to have this feature in general. This is a known fact for eigenvalue crossing. We show that this is also the case for eigenvalues at the threshold of the continuous spectrum by considering the Friedrichs model.
I explain the thermodynamic significance, the duality and open problems associated with the two c... more I explain the thermodynamic significance, the duality and open problems associated with the two colored butterflies shown in figures 1 and 4.
When basic tools of quantum information are applied to the quantum tomography data presented in [... more When basic tools of quantum information are applied to the quantum tomography data presented in [1], none of their devices appears to be a source of entangled photons. In a paper titled “A semiconductor quantum source of triggered entangled photon pairs ” Stevenson et. al. [1] claim to find evidence for the emission of polarization entangled photons from certain quantum dots. A density matrix completely specifies all properties of the quantum state [2]. Using quantum tomography [3], the authors of [1] construct the density matrices representing the polarization state of the photons emitted by each of their devices. We show that subjecting their quantum states to the basic definition of entanglement leads to the inevitable conclusion that none of their devices produced entangled light. In all the dots investigated in [1], quantum tomography yielded real density matrices of the form ρ = α 0 0 γ
We provide a counterexample to the universal paramagnetism conjecture of Hogreve, Schrader and Se... more We provide a counterexample to the universal paramagnetism conjecture of Hogreve, Schrader and Seiler. The counterexample is based on the Bohm—Aharonov effect. Several years ago, one of us [11proved an inequal- (eq. (18) of ref. [41)that ity expressingthe universal diamagnetic tendency of spinless bosons. For a single particle in external (local) Tr(exp(—j3H 2(a, V))) ~ ‘ Tr(exp(—~3H2(a = 0, V))) (6) electric potential V and magnetic potential a, the in- and, in particular, that equality can be expressed
We announce three new rigorous results for the quantum mechanical hydrogen atom in constant magne... more We announce three new rigorous results for the quantum mechanical hydrogen atom in constant magnetic field: (i) Borel summability of the small field perturbation series, (ii) detailed large field asymptotics, and (iii) non-degeneracy of the ground state ti~and a proof that it hasL~t~o = 0 for all values of the field. The weak field Zeeman effect [1] in simple atoms A =-4(r X B); B = (0,0, B) (2) was one of the earliest problems studied [2] in quantum mechanics. More recently, Ruderman [3] and Theorem 1. Let E~(O)be any negative eigenvalue of then others [4] discussed the analogous problem in the HydrogenHamiltonian H(0). Then there is an eigensuper-strong magnetic fields of the type encountered value [10] En(B) of H(B) for B small which is the Borel in neutron stars. It is perhaps surprising that any prob- sum [11,12] of the Rayleigh-Schrödinger perturbation lems remain open for such a well studied theory but coefficients for En [10]. there are some unresolved theoretical questions ...
Abstract. We study the Lyaponov exponent yA(E) of (hu)(n) = u(n + 1) + u(n- l)+AV(n)u(n) in the l... more Abstract. We study the Lyaponov exponent yA(E) of (hu)(n) = u(n + 1) + u(n- l)+AV(n)u(n) in the limit as A +a where V is a suitable random potential. We prove that yA(E)-ln A as A +CO uniformly as E/A runs through compact sets. We also describe a formal expansion (to order A-*) for random and almost periodic potentials. In this note, we study one-dimensional tight binding Hamiltonians h = ho + A V where (hou)(n) = u(n + 1) +u(n- 1). We are interested in the cases where V is either random or almost periodic. By random, we mean that V(n) is a family of identically distributed independent random variables with density P(y)dy where P is bounded with bounded support. In the random case, we will succeed in identifying the first few terms in the large A behaviour of the Lyaponov exponent. For the almost periodic case only a formal large A expansion is obtained. Explicitly, we let yA (E) be the Lyaponov exponent for h, i.e. where 1 yA(E) = lim-lnllM,,(u)...Ml(w)ll n+m n (1) The limit exists...
We study the relation of the adiabatic curvature associated to scattering states and the scatteri... more We study the relation of the adiabatic curvature associated to scattering states and the scattering matrix. We show that there cannot be any formula relating the two locally. However, the rst Chern number, which is proportional to the integral of the curvature, can be computed by integrating a 3-form constructed from the Smatrix. Similar formulas relate higher Chern classes to integrals of higher degree forms constructed from scattering data. We show that level crossings of the on-shell S-matrix can be assigned an index so that the rst Chern number of the scattering states is the sum of the indices. We construct an example which is the natural scattering analog of Berry's spin 1/2 Hamiltonian.
Incommensurate perturbations of classical orbits lead to an almost periodic Hill's operator ... more Incommensurate perturbations of classical orbits lead to an almost periodic Hill's operator whose spectrum, it is argued, is a Cantor set, but one with large Lebesgue measure. Applied to the rings of Saturn, this implies that the complex groove structure in the rings approximates a Cantor set. The possible relevance of the sun in producing 'side gaps' which magnify the apparent gap size is also emphasized
I study the width of the Wannier ladder states, i.e, Bloch electrons in external homogeneous fiel... more I study the width of the Wannier ladder states, i.e, Bloch electrons in external homogeneous field. For periodic potentials with a finite number of gaps, a formula for the width is obtained showing that the width vanishes exponentially fast with the field in accordance with Zener tunneling. The case of infinitely many gaps is studied qualitatively, and it is argued that although the width decreases exponentially "on the average," the detailed bahavior is very complicated. In particular the width oscillates over different orders of magnitude as the field changes slightly. The oscillations are a consequence of a resonance phenomenon. 1. THE PROBLEM * Work supported in part by USNSF MCS-78-01885. I Another possible interpretation of the Hamiltonian is for neutrons in crystals under a gravitational field. A discussion of gravitational interference effects for neutrons is given in Ref. 1281 and references therein. 2 The units are 2m = h = a/2x = 1, a the lattice spacing. The unit of charge is e* = nh*/ma and the unit of force xh*/ma3. ' It is instructive to contrast (I. I) with H = p + V(x)-fx. H is unitarily equivalent top (see footnote (5)) and has no resonances although it has the same symmetry properties as (I. I). The catch is that p + V(x) has no gaps in the spectrum and does not Bragg scatter.
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Papers by J. Avron