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Journal of Mathematical Physics

Schrödinger semigroups for vector fields

T. A. Osborn, R. A. Corns, Y. Fujiwara

Published Mar 1, 1985• 11 citations
Publisher

Abstract

Suppose H is the Hamiltonian that generates time evolution in an N-body, spin-dependent, nonrelativistic quantum system. If r is the total number of independent spin components and the particles move in three dimensions, then the Hamiltonian H is an r×r matrix operator given by the sum of the negative Laplacian −Δx on the (d=3N)-dimensional Euclidean space Rd plus a Hermitian local matrix potential W(x). Uniform higher-order asymptotic expansions are derived for the time-evolution kernel, the heat kernel, and the resolvent kernel. These expansions are, respectively, for short times, high temperatures, and high energies. Explicit formulas for the matrix-valued coefficient functions entering the asymptotic expansions are determined. All the asymptotic expansions are accompanied by bounds for their respective error terms. These results are obtained for the class of potentials defined as the Fourier image of bounded complex-valued matrix measures. This class is suitable for the N-body problem since interactions of this type do not necessarily decrease as ‖x‖→∞. Furthermore, this Fourier image class also contains periodic, almost periodic, and continuous random potentials. The method employed is based upon a constructive series representation of the kernels that define the analytic semigroup {e−zH‖Re z>0}. The asymptotic expansions obtained are valid for all finite coordinate space dimensions d and all finite vector space dimensions r, and are uniform in Rd×Rd. The order of expansion is solely a function of the smoothness properties of the local potential W(x).

DOI
10.1063/1.526631

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Suppose H is the Hamiltonian that generates time evolution in an N-body, spin-dependent, nonrelativistic quantum system. If r is the total number of independent spin components and the particles move in three dimensions, then the Hamiltonian H is an r×r matrix operator given by the sum of the negative Laplacian −Δx on the (d=3N)-dimensional Euclidean space Rd plus a Hermitian local matrix potential W(x). Uniform higher-order asymptotic expansions are derived for the time-evolution kernel, the heat kernel, and the resolvent kernel. These expansions are, respectively, for short times, high temperatures, and high energies. Explicit formulas for the matrix-valued coefficient functions entering the asymptotic expansions are determined. All the asymptotic expansions are accompanied by bounds for their respective error terms. These results are obtained for the class of potentials defined as the Fourier image of bounded complex-valued matrix measures. This class is suitable for the N-body problem since interactions of this type do not necessarily decrease as ‖x‖→∞. Furthermore, this Fourier image class also contains periodic, almost periodic, and continuous random potentials. The method employed is based upon a constructive series representation of the kernels that define the analytic semigroup {e−zH‖Re z>0}. The asymptotic expansions obtained are valid for all finite coordinate space dimensions d and all finite vector space dimensions r, and are uniform in Rd×Rd. The order of expansion is solely a function of the smoothness properties of the local potential W(x).
The asymptotic properties of a Schrödinger wave function which represents the bound ground state of a system of three interacting particles are examined. It is assumed that the interaction can be described by a static potential which is the sum of three two-body potentials and one three-body potential, where the potentials have the property that if any one of the particles is separated from the other two by a distance which tends to infinity, then the part of the potential energy which depends on the position of that particle tends to zero. The problem is treated nonrelativistically. Decreasing exponential bounds on the ground-state three-body wave function are established in configuration space. It is shown that these bounds depend only on the masses of the three particles, on the ground-state energy of the three-body system, and on the lower bounds on the spectra of the Hamiltonians for the three two-body systems arising when one of the three particles is removed and the remaining two interact through the corresponding two-body potential. It is furthermore shown that the three-body wave function in momentum space admits of analytic continuation into a tube-region in complex momentum space, which tube-region is the product of real momentum space and a convex bounded region in imaginary momentum space. The tube-region is explicitly determined. The implications of the results for the theory of the vertex function in quantum field theory are discussed. The relevancy of the results obtained in this paper to the variational computation of energy levels and wave functions of three-body systems is also briefly discussed. In the course of the derivation some generalizations are pointed out; in particular, the final results remain valid for a class of more general interactions, which cannot be described by static potentials. Likewise, the results remain valid for a class of spin-dependent interactions.
The quantum evolution of an N-body system of particles that mutually interact through scalar fields and couple to an arbitrary external electromagnetic field is rigorously described. Both operator and kernel valued solutions to the evolution problem are found. Based upon a particular realization of the Dyson expansion, a convergent series representation of the propagator (the kernel of the Schrödinger time evolution operator) is obtained. The basic approach is to embed the quantum evolution problem in the larger class of evolution problems that result if mass is allowed to be complex. Quantum evolution with real mass is considered to be the boundary value of the complex mass evolution problem. The constructive representation of the propagator is determined for the class of analytic scalar and vector fields that are given as Fourier transforms of time-dependent scalar and vector-valued measures.
The time evolution operator U(t, s) of a spinless nonrelativistic N-body quantum system in Euclidean space with real analytic time-dependent scalar interaction v(x, t) is studied. A complete formal asymptotic expansion involving simple connected graphs is derived for the full coordinate, and mixed coordinate-momentum representation propagators. The derivation is based on Dyson's series for U(t, s), and the combinatorics involved in the cluster expansion of the classical grand partition function. These results provide an efficient means of generating nonperturbative propagator expansions in the physical variables: mass m, Planck's constant h, time displacement t-s. The structural bridge between the WKB and Wigner-Kirkwood expansions is sketched for mixed representations of U(t, s). In the heat equation context the graphical expansions are found for mixed representations of the density operator e - beta H . Finally, an explicit differential formula is obtained for Wigner's distribution function f(x, p; beta ).
We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on Rn{\mathbb{R}^n}Rn , with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in ℏ{\hbar}ℏ , given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the t→0{t \to 0}t→0 limit approaches the δ distribution. As such, our construction gives explicitly the full ℏ→0{\hbar\to 0}ℏ→0 asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.
Linear combination of Hamiltonian simulation (LCHS) provides an asymptotically optimal framework for the quantum simulation of linear non-unitary dynamics. The LCHS approach approximately implements the Peano--Baker propagator through an integral representation determined by a kernel function. We establish an LCHS approximation theorem for a broadened class of admissible kernel functions, defined by natural complex-analytic conditions. In the standard time-independent block-encoding access model, we explicitly construct, for each target error ε\varepsilonε, the unique kernel function in this class minimizing the query-cost functional and show that the minimum of this functional has the asymptotic expansion 2eπ(log⁡1ε−log⁡log⁡1ε+o(1))\tfrac{2\mathrm{e}}π\big(\log\frac{1}{\varepsilon}-\log\log\frac{1}{\varepsilon}+o(1)\big)π2e​(logε1​−loglogε1​+o(1)) as ε↓0{\varepsilon\downarrow0}ε↓0. The provably optimal leading coefficient 2e/π2\mathrm{e}/π2e/π improves by a factor of approximately 2.1 on the coefficient obtained empirically by Low and Somma through numerical minimization over their family of optimally scaling kernel functions. We construct a spectrally convergent quadrature with explicit bounds on operator error and linear combination of unitaries (LCU) subnormalization, proving that this asymptotic improvement persists after discretization into a finite linear combination of unitaries suitable for implementation. Our results are based on Hardy space theory and convex analysis.
Connected graph methods are used to solve the transport equation arising in a semiclassical representation of the Schrodinger propagator K(t,s,x,y). For a spinless quantum system whose Hamiltonian is a smooth position- and momentum-dependent perturbation of the Laplacian in R d , exponential cluster expansions for K are obtained. These formally exact solutions to the time-dependent Schrodinger equation employ explicit graphically-determined derivatives of the Hamiltonian's symbol, integrated along geodesics. For the special case of particles interacting with external electromagnetic fields, the propagator's gauge invariant derivative expansion coefficients are determined in closed form in terms of the Lorentz force. A tree graph formula for Hamilton's principal function is extracted from this result.