Unbounded random operators and Feynman formulae

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Feb 11, 2016 - averaging (with the help of Feynman formulae) the random one- ... or Hamiltonian Feynman formulae respectively (there are Feynman formulae ...
Izvestiya: Mathematics 80:6 1131–1158

Izvestiya RAN : Ser. Mat. 80:6 141–172

DOI: https://doi.org/10.1070/IM8402

Unbounded random operators and Feynman formulae Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov Abstract. We introduce and study probabilistic interpolations of various quantization methods. To do this, we develop a method for finding the expectations of unbounded random operators on a Hilbert space by averaging (with the help of Feynman formulae) the random one-parameter semigroups generated by these operators (the usual method for finding the expectations of bounded random operators is generally inapplicable to unbounded ones). Although the averaging of families of semigroups generates a function that need not possess the semigroup property, the Chernoff iterates of this function approximate a certain semigroup, whose generator is taken for the expectation of the original random operator. In the case of bounded random operators, this expectation coincides with the ordinary one. Keywords: quantization, one-parameter semigroup, random operator, Hamiltonian operator, Hamiltonian function, Chernoff’s formula, Feynman formula, Chernoff equivalence, randomization, probabilistic interpolation.

Feynman formulae (see [1]) express the Schr¨ odinger semigroup exp(−tH), t > 0, or the Schr¨odinger group exp(itH), t ∈ R, in terms of limits of integrals over the Cartesian powers (with exponent tending to infinity) of a certain space related to the classical Hamiltonian system whose quantization gives the Hamiltonian operator H. In particular, H may be a pseudo-differential operator whose symbol is the classical Hamiltonian function H. When the integrals are taken over the Cartesian powers of the configuration space or the phase space, one speaks of Lagrangian or Hamiltonian Feynman formulae respectively (there are Feynman formulae not belonging to either of these types). In this paper we consider only Lagrangian Feynman formulae. Feynman was the first to suggest representing the Schr¨odinger group or semigroup as a limit of integrals over the Cartesian powers of the configuration space or phase space. He considered the case of the configuration space in 1948 [2] and that of the phase space in 1951 [3]. The integrands in his Lagrangian and Hamiltonian formulae (see [3], [1]) were exponentials of approximants to the action functional. The first proof of a formalized version of Feynman’s results of 1948 was given in 1964 by Nelson, who used Trotter’s formula (see [4]). A proof of Feynman’s results O. G. Smolyanov was supported by RFBR (grant no. 14-01-00516). The work of V. Zh. Sakbaev was supported by the Russian Science Foundation (grant no. 14-11-00687) at the Steklov Mathematical Institute of the Russian Academy of Sciences. AMS 2010 Mathematics Subject Classification. 46G10, 47D08, 81Q30.

c 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

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of 1951 was obtained only in 2002 in [1] using Chernoff’s formula (see [5]), which had already been published in 1968. After the appearance of [1] it became clear that Chernoff formulae provide an effective method of obtaining approximants for the groups and semigroups arising in quantum mechanics and stochastic analysis. Multiple integrals in Feynman formula are approximants to the integral over the infinite-dimensional space in the Feynman–Kac formula (see [6]). These formulae can be used (see [7]) to represent the quantum evolution operators e−itH and the equilibrium density operators e−βH in the study of random semigroups and random Hamiltonians, in the search for representations of solutions of evolutionary differential equations with variable coefficients (in particular, such equations describe the quantum dynamics or diffusion of particles whose mass depends on the coordinate and momentum), in the study of Schr¨odinger-type stochastic differential equations and quantum stochastic differential equations, in the investigation of surface measures on submanifolds of infinite dimension and codimension in infinite-dimensional spaces (see, for example, [8]), in the search for representations of solutions of equations on functions on Riemannian manifolds, and in many other problems of mathematical physics, stochastic analysis and mathematical biology. A unified approach to these problems is to construct an appropriate (possibly random) operator-valued function F(t), t > 0, which generally does not possess the semigroup property, and then define a sequence {Gn (t), t > 0} of operator-valued functions by the formula   n t Gn (t) = F , t > 0. n The limit of the sequence Gn is a semigroup that solves the corresponding problem. This paper is devoted to a discussion of constructions connected with such problems. One of our motivations was an application of these constructions to obtain what we call the probabilistic interpolation of various quantization methods (see Definition 2 below). This interpolation can be applied in cases when there is no natural single-valued map of the objects describing the classical dynamics of the system to the quantum operators (such a map is usually referred to as a quantization). § 1. Preliminaries We shall make essential use of the following theorem (see [5] and [9]). Let X be a Banach space, B(X) the Banach space of bounded linear operators on X endowed with the strong operator topology, and F : [0, +∞) → B(X) a function with F(0) = I which is continuous in the strong operator topology on B(X) and satisfies kF(t)kB(X) 6 eαt , t > 0, for some α > 0. Then one can find a dense vector subspace D of X such that for every u ∈ D the limit lim t−1 (F(t)u − u) ≡ F0 (0)u

t→+0

exists. If the operator F0 (0) with domain D is closable and its closure is the generator of a strongly continuous operator semigroup U(t), t > 0, then for every u ∈ X and every T > 0 we have

  n

t

lim sup U(t)u − F u

= 0. n→∞ t∈[0,T ] n X

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Such approximations of one-parameter semigroups by iterates of operator-valued functions will be the object of our study. They are called Chernoff product formulae and may become Feynman formulae when written for integral operators F(t). Feynman formulae appear in our paper in precisely this way and are used to obtain analogues of expectation for unbounded random operators. The corresponding operation is referred to as the averaging of these operators. We shall also see how to use this procedure to quantize classical Hamiltonian systems. We notice that the averaging (extended by linearity) of self-adjoint operators may give only symmetric and not self-adjoint operators (Example 2). This is related to Naimark’s theorem that every symmetric operator is generated (see [10]) by a self-adjoint operator acting on a larger Hilbert space (see [11]). Here one can assume that these symmetric operators describe the dissipative dynamics generated by the Schr¨odinger dynamics in the larger space. The following representation of the Schr¨ odinger semigroup by a Hamiltonian Feynman formula was established in [1]:   n t exp(−tH) = lim f , n→∞ n

(1)

where f (t) stands for the pseudo-differential operator whose classical symbol is the function [(q, p) → exp(−tH(p, q))], (p, q) ∈ R2d (the operator f (t) is referred to as the quantization of this symbol), and the limit is taken in the strong operator topology (as in Chernoff’s theorem). The formula (1) was used in [12] to obtain a representation of the semigroup exp(−tH), t > 0, generated by the oscillator Hamiltonian. Although the quantum oscillator Hamiltonian H is independent of the choice  of quantization, the quantization f (t) of the symbol (q, p) → exp − 2t (q 2 + p2 ) is an operator depending non-trivially on this choice. The dependence of Feynman formulae for the semigroup exp(−tH), t > 0, on the way of quantizing this symbol was discussed in [12] (of course, the semigroup itself is independent of quantization).1 The main objects to be considered in this paper are the evolutionary differential equations d u(t) = Lε u(t), t > 0, (2) dt with random coefficients and the corresponding (random) semigroups, where the random variables are understood in the following extended sense. By a measurable space (Ω, A) we mean an arbitrary set Ω (called the set of elements of the measurable space) endowed with an algebra A of subsets (called measurable subsets). A map ξ : (Ω1 , A1 ) → (Ω2 , A2 ) between measurable spaces is said to be measurable if the full pre-image of every measurable set (in A2 ) is measurable (in A1 ). When the algebra in question is clear, we sometimes say 1 We stress that a direct computation of the limit (1) is technically difficult even for the simplest Hamiltonian of a quantum harmonic oscillator. Berezin [13] regarded the formula (1) for the Hamiltonian H = 12 (q 2 + p2 ) as obvious and did not even pose the problem of estimating the rate of convergence in the Feynman formula and studying the dependence of this rate on the choice of quantization. In [12] we obtained an exact expression for the principal part of the deviation of the limit in (1) from its finite approximants and constructed Wigner functions corresponding to the finite approximants of the equilibrium density operator for the Weyl quantization.

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‘measurable space Ω’ instead of ‘measurable space (Ω, A)’. We endow every topological space with the structure of a measurable space by letting the algebra of measurable sets be the minimal algebra containing all open subsets. Every subset of a topological space is endowed with the induced topology and, therefore, with the corresponding structure of a measurable space. A probability measure on a measurable space (Ω, A) is a non-negative normalized finitely additive function µ on the algebra A. A probability space (Ω, A, µ) is a measurable space (Ω, A) endowed with a probability measure µ on the algebra A. We sometimes replace the symbol (Ω, A, µ) by (Ω, µ). By a random variable we mean a measurable function ξ on a probability space (Ω, A, µ) with values in some measurable space. If the random variable ξ takes values in the space of operators, it is called a random operator. The notion of a random semigroup is defined in a similar way (see Definition 1 below). In what follows we assume that the symbol Lε on the right-hand side of (2) means a differential operator acting on some Banach function space X. The coefficients and the domain of this operator may depend on ε ∈ Ω. The operator Lε is also assumed to be the generator of a strongly continuous semigroup Uε of transformations of X. We suggest an approach that naturally extends the procedure of averaging of bounded linear operators to unbounded ones. Averaging of random operators is called a probabilistic interpolation (Definition 2). Similar terminology is used for the random semigroups generated by these operators. Such semigroups are defined as follows. Let Ys (X) be the topological vector space of strongly continuous maps F of the semi-axis R+ = [0, +∞) to the Banach space B(X) of linear transformations of a Banach space X. The topology τs on Ys (X) is determined by the family of functionals φT,u , T > 0, u ∈ X, acting by the rule φT,u (F) = supt∈[0,T ] kF(t)ukX . We also assume that there is a Banach space X∗ such that (X∗ )∗ = X. Definition 1. A random semigroup is a random variable G with values in the set S(X) ⊂ Ys (X) of strongly continuous one-parameter semigroups of operators on a Banach space X. The expectation FG of a random semigroup G will be defined (by the formula (3) below) as an operator-valued function which need not be a semigroup in general. Definition 2. A semigroup UG is called a generalized mean value (or a probabilistic interpolation) of a random semigroup G if the expectation FG of G is Chernoff equivalent to UG (the Chernoff equivalence [14] of an operator-valued function FG  and a semigroup UG means that the sequence of operator-valued functions  n FG nt , t > 0 converges as n → ∞ in the strong operator topology to the semigroup UG (t), τ > 0, uniformly on every interval [0, T ] of the half-line t > 0).2 We now describe the notion of the generator of a random semigroup. Transfer the structure of a measurable space to the set G(X) of generators of all semigroups 2 Linear quantization of the exponential function of a classical Hamiltonian (see [15], [16] and the references therein) and averaging of semigroups generated by random quantum Hamiltonians give rise to operator-valued functions which are not semigroups. But Feynman’s formula (that is, Chernoff’s theorem) generates a semigroup in both cases. In the first case, its generator is an appropriate quantization of the classical Hamiltonian. In the second, it is the mean value of the random operator obtained by randomizing (see below) a family of Hamiltonians (this family arises because of the non-uniqueness of quantization).

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in S(X). Let G be a random semigroup on a probability space (Ω, A, µ). Then its generator is the random variable HG (on the same probability space) whose value HG (ε) at every ε ∈ Ω is the generator of the semigroup G(ε) ∈ S(X). The topology τG on the set G(X) is defined by requiring that the bijection J between S(X) and G(X) (sending each semigroup to its generator) is a homeomorphism of the topological spaces (S(X), τS ) and (G(X), τG ). Thus HG = J ◦ G. In other words, HG is a measurable function on the probability space (Ω, A, µ) with values in the topological space G(X) of generators. If the expectation of the random semigroup G is an operator-valued function FG which is Chernoff equivalent to some semigroup UG , then the generator of UG is called the expectation of the random generator HG of the random semigroup G. This definition of averaging of generators may be regarded as an extension of the procedure of averaging in the space of bounded operators since in the case when all values of the random generator are bounded, its mean value may coincide (see Corollary 2 of Theorem 2) with the ordinary mean value of elements of a Banach space. In Examples 1, 2 (§§ 3, 5 respectively), when the values of the random generator are unbounded operators, the natural definition of the expectation of HG can give the zero domain or lead to operators which are not generators, even after taking their closures. But our definition works in both cases and produces the generator of a semigroup. Note that in Example 2, even averaging by means of quadratic forms followed by closure does not give rise to a generator. The following definitions formalize the idea of combining various methods of quantization. This formalization contains no recipe for choosing the most appropriate combination: such a choice must be determined by the physical situation studied. Definition 3. A randomization (or µ-randomization) of a subset F of a topological space (Y, τ ) is a probability measure µ on (the measurable space) F. When (Y, τ ) is a topological vector space, one can define the mean value of µ. It coincides with the expectation of the (Y, τ )-valued random variable F 7→ F on the probability space (F, µ). Definition 4. Let µ be a randomization of a subset F of the topological vector space Ys (X). A strongly continuous semigroup Uµ of bounded linear operators on the Banach space X is called a (probabilistic) µ-interpolation of F if it is Chernoff equivalent to the mean value of µ. Thus, a strongly continuous semigroup Uµ of bounded linear operators on X is a µ-interpolation of F if for every T > 0 and every u ∈ X we have

  n

µ t u lim sup U (t)u − Fµ

= 0, n→∞ t∈[0,T ] n X where the operator-valued function Fµ is the mean value of µ. By a quantization of functions on the phase space Q × P we mean a map Ψ from a set H(Q, P ) consisting of some real-valued functions on Q × P to the set G(H) of self-adjoint operators on a separable Hilbert space H. For every h ∈ H(Q, P ) the operator Ψ(h) is referred to as the quantization of h.

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Let M be a set of quantizations of functions on the phase space Q × P . Definition 5. Suppose that h ∈ H(Q, P ), µ is a randomization of the set −iM(h) = {−iΨ(h) : Ψ ∈ M} and µ0 is the following randomization of the set U of all unitary groups generated by elements of −iM(h): µ0 is the image of µ under the map sending each generator to the group it generates. Then the product of the imaginary unit and the generator of the unitary group obtained as the µ0 -probabilistic interpolation of U is called the interpolation (or, more precisely, the µ-interpolation) of quantizations M of the function h. Note that in the physics literature the generator of a unitary Schr¨odinger group (e−itH )t∈R is understood to be the self-adjoint operator H, but we use the terminology of operator semigroups where the generator of this group is −iH. § 2. Random semigroups associated with the Cauchy problem, and Chernoff equivalence Let (E, A) be a measurable space and W (E, A) the set of non-negative normalized finitely additive measures on it. We regard a random semigroup as a Zs -valued or Zw -valued random variable ξ on the probability space (E, A, µ) for some measure µ ∈ W (E, A), where Zs = Cs (R+ , B(X)) (resp. Zw = Cw (R+ , B(X))) is the topological space of strongly (resp. weakly) continuous maps from the semi-axis R+ to the Banach space B(X) of bounded linear transformations of a Banach space X dual to the Banach space X∗ . On every space Z ∈ {Zs , Zw } we define a family of functionals acting on any z ∈ Z by the rule ϕt,A,g (z) = hz(t)A, gi, t ∈ R+ , A ∈ X, g ∈ X∗ . Consider two topologies on Z: the topology τZ,w generated by the family of functionals ϕt,A,g , t ∈ R+ , A ∈ X, g ∈ X∗ , and the topology τZ,s generated by the family of functionals Φt,A , t ∈ R+ , A ∈ X, where Φt,A (z) = kz(t)Ak. Let Aτ be the minimal algebra containing the topology τ ∈ {τZ,s , τZ,w }. Then Z endowed with the algebra Aτ is a measurable space and the map ξ : E → Z is a random variable. By the expectation of a random variable ξ (a measurable map from the measurable space (E, A, µ) to the measurable space (Z, Aτ )) we mean the Pettis integral Z Mξ = ξε dµ(ε), E

where M ξ is an element of Z such that for all t ∈ R+ , A ∈ X, g ∈ X∗ we have Z hM ξ(t)A, gi = hξε (t)A, gi dµ(ε). (3) E

Here hA, gi stands for the value of a continuous linear functional A ∈ X = X∗∗ on the element g ∈ X∗ . The following theorem gives sufficient conditions for the existence of (3) as the integral of a complex-valued function with respect to a finitely additive measure µ ∈ W (E) (see [17]).

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Theorem 1. Let µ be a real-valued measure of bounded variation on an algebra A of subsets of E. If ξ : E → Z is a measurable map with bounded image and there is a dense vector subspace D of X such that for every A ∈ D the family of maps ξε (t)A ∈ C(R+ , X), ε ∈ E, is weakly (resp. strongly ) uniformly Lipschitz, then M ξ(t) ∈ Cw (R+ , B(X)) (resp. M ξ(t) ∈ Cs (R+ , B(X))). Proof. The dense weak uniform Lipschitz property of ξ means that for all A ∈ D and g ∈ X∗ there is a constant L > 0 such that sup

|hξε (t + ∆t)A − ξε (t)A, gi| 6 L|∆t|,

t∈R+ , ε∈E

and the strong one means that for every A ∈ D there is a constant L > 0 with sup

kξε (t + ∆t)A − ξε (t)AkX 6 L|∆t|.

t∈R+ , ε∈E

The uniform boundedness of ξ means that kξε (t)kB(X) 6 C

sup ε∈E, t∈R+

for some C > 0. Since ξ is measurable and uniformly bounded, it follows that hξε (t)A, gi is a bounded measurable complex-valued function on (E, A) for every t > 0 and all A ∈ X and g ∈ X∗ . Hence this function is integrable with respect to the finitely additive measure µ of bounded variation (see [17], § 4.9, Example 5). Moreover, as a function of g, the integral (3) is a continuous linear functional on X∗ . Hence for every A ∈ X we have a Pettis integral Z ξε A dµ(ε) ∈ X E

and the map Z A→

ξε A dµ(ε) E

is linear in A (since the Pettis integral is linear) and continuous (since ξ is uniformly bounded). Hence the map Z A→

ξε A dµ(ε) E

is defined on X and is a bounded linear transformation of X. Thus for every t > 0 there is a mean value Z M ξ(t) = ξε (t) dµ(ε) ∈ B(X). E

The vector-valued function M ξ(t)A of the variable t ∈ R+ is weakly (strongly) continuous and even weakly (strongly) Lipschitz for all A ∈ D since the inequality expressing the weak (strong) Lipschitz property of ξ is preserved under integration, up to multiplying the Lipschitz constant by the variation of the measure. For

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every u ∈ X there is an A ∈ D such that ku − AkX 6 σ. Hence, by the uniform boundedness, we have kM ξ(t)u − M ξ(t)AkX 6 Cσ for all t > 0. It follows that the vector-valued function M ξ(t)u of the variable t ∈ R+ is weakly (strongly) continuous for all u ∈ X.  Remark. This construction of the expectation of a random map ξ : E → Cw (R+ , B(X)) uses the Pettis integral and the space X (which is dual to the Banach space X∗ ) to guarantee that the expectation M ξ is an element of the same space Cw (R+ , B(X)). For an arbitrary Banach space X one can define the expectation of a random map ξ using the Gel’fand integral as an element of Cw (R+ , B(X ∗∗ )). Thus under the hypotheses of Theorem 1 the expectation of a random semigroup ξ is again an element of Cw (R+ , B(X)) or Cs (R+ , B(X)). This element is referred to as the averaging (or the mean value) of the family ξ of transformations of X. By the theorem on bounded perturbations of generators of strongly continuous semigroups ([18], Theorem 9.2.1), the mean value of a random semigroup depends continuously on uniformly bounded (in the operator norm) perturbations of the random generator. We now give examples of the use of random semigroups. 1. One can use random semigroups to regularize Cauchy problems whose wellposedness is unknown or absent. For example, approximating a maximal symmetric operator (not generating a semigroup) by sequences of generators was studied in [19]. The non-uniqueness of the choice of approximants and the absence of convergence of the sequence of regularized semigroups can then be compensated by randomization. 2. Random semigroups related to a quantum system can be used in connection with the non-uniqueness of a classical Hamiltonian describing a family of trajectories in the classical coordinate space (see [20]). Then the space of elementary events is the set K of equivalent classical Hamiltonians corresponding to the given law of motion in the coordinate space Q, along with a non-negative normalized measure µ on the algebra 2K of all subsets of K. Quantization of classical systems can be regarded as a map from the set of classical Hamiltonians to the set of quantum Hamiltonians (self-adjoint operators on the Hilbert space L2 (Q)) or further, to the set of unitary groups generated by the quantum Hamiltonians on L2 (Q). Hence it is a random variable with values in the set of self-adjoint operators (or unitary semigroups). However, we stress that when using this method of averaging of random semigroups for quantization purposes, the corresponding probability measure may be regarded as an additional parameter that determines the method of quantization. We do not discuss the choice of this parameter here. As already mentioned, the expectation of a random semigroup need not be a semigroup. The following elementary observation gives an explicit example of losing the semigroup property under averaging of semigroups. The complex-valued functions eit , t > 0, and e−it , t > 0, are strongly continuous unitary semigroups of transformations of the one-dimensional Banach space X = C, but their semi-sum

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+ e−it ) = cos t is not a semigroup of transformations of X since cos(t + s) = cos t cos s − sin t sin s and the semigroup property cos(t + s) = cos t cos s does not hold. But even operator-valued functions that have lost the semigroup property can contain information on a certain semigroup, which is uniquely determined by the operator-valued function using the iteration procedure used to construct the Feynman approximants of semigroups on the basis of Chernoff’s theorem. Following [14], [15], we introduce an equivalence relation on the set Π of strongly continuous operator-valued functions F : [0, +∞) → B(X) such that F (0) = I, kF (t)kB(X) 6 1 + ct for some c > 0, t ∈ [0, δ), and there is a derivative F 0 (0) whose closure is the generator of a strongly continuous semigroup. 1 it 2 (e

Definition 6. The operator-valued functions F, G ∈ Cs (R+ , B(X)) with F(0) = G(0) = I are said to be Chernoff equivalent if for every T > 0 and every u ∈ X we have

  n   n 

t t − F u lim sup G

= 0. n→∞ t∈[0,T ] n n § 3. Applications of averaged families of transformations to quantization of Hamiltonian systems of classical mechanics Let H = H(q, p), (q, p) ∈ R2d , be a twice continuously differentiable Hamilton function of an autonomous classical mechanical system with phase space R2d = Rd × Rd (with the standard symplectic form) and coordinate space Q = Rd . By ∇q H (resp. ∇p H) we understand the d-tuple of partial derivatives with respect to the first (resp. last) d real arguments of H. When a pair of vector-valued functions f = f (t), g = g(t), t ∈ R, satisfies the system of Hamilton equations f˙(t) = ∇p H(f (t), g(t)),

g(t) ˙ = −∇q H(f (t), g(t)),

(H)

the function f is called the configuration trajectory (or trajectory in Q) of the solution (f, g) of this system. We put ΓH = {f : ∃g, (f, g) : R → R2d is a solution of (H)}. This is the set of configuration trajectories of the system (H) in Q. By a classical Hamiltonian we mean a twice continuously differentiable function on the phase space R2d which is strictly convex in the variables p ∈ Rd . In [20], it was suggested that we regard Hamiltonian systems on R2d with classical Hamiltonians H 0 and H as equivalent if ΓH 0 = ΓH , that is, every configuration trajectory of the system with Hamiltonian H 0 is a configuration trajectory of the system with Hamiltonian H, and vice versa. This is indeed an equivalence relation on the set of Hamiltonians on R2d . Examples of equivalent Hamiltonians are the p Hamiltonians of free motion: H(q, p) = e p) = p2 /2 + p4 /4, H 0 (q, p) = 1 + p2 and so on. p2 /2, H(q, Although distinct equivalent Hamiltonians (or Lagrangians) determine the same set of trajectories in the coordinate space of an isolated classical system, they give rise to different quantum dynamics. When the classical system is not isolated, the

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set of trajectories in the configuration can already depend on the choice between equivalent Hamiltonians (or Lagrangians). Let M be the set of classical Hamiltonians equivalent to a Hamiltonian H. Then the sets ΓH of the configuration trajectories corresponding to distinct Hamiltonians H ∈ M are indistinguishable (but the trajectories in the phase space can be different, and the quantum semigroups generated by the quantum Hamiltonians corresponding to distinct equivalent classical Hamiltonian are, generally speaking, distinct). An example of a random semigroup is a map from the set M (endowed with a measure ν) of equivalent classical Hamiltonians to the vector space of strongly continuous semigroups on the Hilbert space H = L2 (Rd ). Let µ be a probability measure supported on some countable subset {Ln } ⊂ G(X), where G(X) is the set of generators P∞of contraction semigroups in a Banach space X. Then µ(Ln ) = µn > 0 and n=1 µn = 1. The formal mean value of R this sequence of operators with respect to µ is the integral G(X) Ln dµ or, more P∞ concretely, the operator series n=1 Ln µn . Our purpose is to make sense of this series. Notice that even the partial sums Sm =

m X

Ln µn ,

m ∈ N,

n=1

P∞ of the series n=1 Ln µn need to be defined. Consider the dynamical properties of the mean value Z X F(t) = e−itLn dµ = e−itLn µn , M

t > 0,

n∈N

of the corresponding random semigroup. As mentioned above, the mean value of a random semigroup need not be a semigroup. But the following theorem shows that the mean value F(t), t > 0, of the random semigroup generates a semigroup Chernoff equivalent to F(t). The generator of this semigroup is the mean value of the Hamiltonian. Theorem 2. Let {Ln } be a sequence of generators of strongly continuous semigroups in a Banach space X, {µn } a sequence of non-negative numbers with total sum 1, and D ⊂ X a vector subspace P such that D is an essential domain for every ∞ operator Ln , n ∈ N, and the series converges for every x ∈ D. n=1 µn kLn xk P∞ Define an operator S on D by the formula Sx = k=1 µk Lk x. If S is closable and its closure generates a strongly continuous semigroup U(t) = etS , t > 0, then P tLn the expectation of the random semigroup F(t) = n∈N e µn , t > 0, is Chernoff equivalent to the semigroup etS , t ∈ R+ . Remark. The measure µ on the set of values of the sequence {Ln } is a randomization of this set and the operator S is a probabilistic interpolation of this randomization. The randomization makes the sequence {etLn , t > 0} into a random semigroup whose generalized mean value is the semigroup etS , t > 0. P∞ Proof. By the hypotheses of Theorem 2, the series k=1 µk Lk x converges to Sx in the norm topology of X. Thus the limiting linear operator S : D → X is defined on the vector space D.

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Let us verify that all hypotheses of Chernoff’s theorem hold for the function F. Since e−itLn |t=0 = I for all n ∈ N and µ(N) = 1, we have F(0) = I. Take u ∈ X. Since the contraction semigroups Un (t) = etLn , t ∈ R+ , are strongly continuous, the functions Un (t)u, t ∈ R+ , are continuous for every n ∈ N and uniformly bounded by the constant kukX . Since ke−tLn kB(X) 6 1 for all n ∈ N and µ is a non-negative normalized measure on N, the Weierstrass test P yields that the series n∈N etLn uµn converges uniformly with respect to t ∈ R+ to the continuous function F(t)u, t ∈ R+ . Hence the operator-valued function F(t), t ∈ R+ , is strongly continuous. Moreover, X kF(t)kB(X) 6 ketLn kB(X) µn = µ(N) = 1. n∈N

Hence we have kF(t)kB(H) 6 1. P We now establish that for every u ∈ D the function F(t)u = k∈N µk Uk (t)u is continuously differentiable on [0, +∞) and find the limiting value F0 (+0)u of the derivative. tLk Fix u ∈ D. Then the function vk (t) = Pe u, t ∈ R, is continuously differentiable on R+ for every k ∈ N and the series k∈N µk vk (t) converges uniformly on [0, T ] for every T > 0 to a continuous function v(t). We claim that for every u ∈ D the function v(t) = F(t)u is differentiable at zero and the derivative is equal to Su. Indeed, by hypothesis, X k∈N

µk

X X d vk (t) = µk Lk vk (t) = µk Lk etLk u. dt k∈N

k∈N

The series k∈N µk Lk etLk u converges uniformly on the semi-axis R+ by the Weierstrass test and the hypotheses of the theorem: X

X tL

µk Lk etLk u 6 µk e k Lk u P

k∈N

k∈N

and, since we are dealing with contraction semigroups, X

X

µk Lk etLk u 6 µk kLk uk < ∞ k∈N

k∈N

for every u ∈ D. It follows that for every u ∈ D the function F(t)u, t > 0, is continuously differentiable on [0, +∞), and its derivative tends to X n∈N

µn

d un (+0) = Su dt

as t → +0. Hence all hypotheses of Chernoff’s theorem hold for the function F(t), t > 0, and

  n 

t −itS lim F − e u

= 0. n→+∞ n H Thus F is Chernoff equivalent to the semigroup U. 

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Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov

Theorem 2 is of interest even when the series is finite. It is an additive analogue of Trotter’s theorem. We P also note that in our proof one cannot allow negative µn , even when the ∞ series n=1 µn converges absolutely. This is because the operator norm of F (t) can then be unbounded for small t. Indeed, one can find unbounded self-adjoint operators A1 , A2 on a Hilbert space H and numbers µ1 , µ2 ∈ R with µ1 + µ2 = 1 such that for every t > 0 and every δ > 0 there is a unit vector u ∈ H satisfying

(µ1 e−itA1 + µ2 e−itA2 )u > |µ1 | + |µ2 | − δ. H For the operators A1,2 one can take commuting operators that have an orthonormal (1,2) basis of eigenvectors {ek } with eigenvalues {λk } respectively. Then  (1) (2)  µ1 e−itA1 + µ2 e−itA2 ek = µ1 e−itλk + µ2 e−itλk ek and, therefore,

 (2) (1)

µ1 e−itA1 + µ2 e−itA2 ek = µ1 + µ2 e−it(λk −λk ) . H (1,2)

(2)

(1)

The eigenvalues {λk } can be chosen in such a way that the sequence λk − λk , k ∈ N, is dense in R. It follows that the function F(t) = µ1 e−itA1 + µ2 e−itA2 does not satisfy the hypothesis kF(t)k 6 eat , t ∈ (0, δ), of Chernoff’s theorem. The last hypothesis of Theorem 2 (the existence of a common essential domain for the terms of the operator series and its partial sums) is rather cumbersome and not easily verifiable. The following assertion gives a sufficient condition for it to hold. Corollary 1. P Suppose that L1 , . . . , Lm are quantum Hamiltonians on a Hilbert m space X and j=1 pj = 1, pj > 0. If the quadratic form of the operator L1 majorizes the quadratic forms of the operators Lk , k = 2, . . . , m (see [18]), then the domain D(L1 )Pis an essential domain for the other operators and for the mean m Hamiltonian L = j=1 pj Lj . Moreover, the mean value of the semigroups is Chernoff equivalent to the semigroup generated by the mean Hamiltonian. Corollary 2. If {Lj } is a uniformly bounded sequence of operators in B(X), then the conclusion of Theorem 2 holds. Definition 7. a) A maximal dissipative operator L is called the sum of maximal dissipative operators p1 L1 and p2 L2 for arbitrary p1 > 0, p2 > 0, p1 +p2 = 1, in the exponential sense if the operator-valued function G(t) = p1 etL1 + p2 etL2 ,

t > 0,

is Chernoff equivalent to the semigroup etL . b) A maximal dissipative operator L is called an integral of a function L(ε), ε ∈ E, with values in the set of maximal dissipative operators, with respect to a non-negative normalized measure µ on the algebra 2E in the exponential sense if the operator-valued function Z G(t) = etL(ε) dµ(ε), t>0 E

Unbounded random operators and Feynman formulae

1143

(where the integral is understood in the Pettis sense) is Chernoff equivalent to the semigroup etL . By Corollary 2, the averaging procedure (defined above) for generators of semigroups generalizes the averaging procedure for bounded linear operators. Clearly, the sum of operators is commutative and associative and the zero operator is a zero element. Theorem 3 is proved similarly to Theorem 2. Theorem 3. Let {Lε , ε ∈ E} be an operator-valued function from a set E to the set of generators of strongly continuous contraction semigroups on a Banach space X, µ a non-negative normalized countably additive measure on the σ-algebra 2E of all subsets of E, and D ⊂ X a vector R subspace such that D is an essential domain for each operator Lε , ε ∈ E, and E kLε xk dµ(ε) converges for all x ∈ D. Define an operator S on D by the formula Z Sx = Lε x dµ. E

If S is closable and its closure generates a strongly continuous semigroup U(t) = etS , t > 0, then the mean value of the random semigroup Z F(t) = etLn µn , t > 0, E

is Chernoff equivalent to the semigroup etS , t ∈ R+ . Example 1. This example shows that the average of two generators of semigroups in the sense of Definition 7 can be densely defined and coincide with the sum of unbounded operators in the sense of quadratic forms although the intersection of the domains of these operators (that is, the domain of a linear combination of them) is the zero subspace. We recall that in the more complicated Example 2 below (§ 5) our averaging method is even strictly stronger than the method of quadratic forms. d2 On the Hilbert space H = L2 (R) we define an operator A = − dx 2 with domain   d d D(A) = W22 (R) and an operator B = − dx g dx with domain D(B) = u ∈ W21 (R): d u ∈ W21 (R) . g dx Here the function g ∈ L∞ (R) satisfies 1 6 g(x) 6 2, x ∈ R, and is defined as follows. Let {rk } be a sequence exhausting all rational numbers. Put X g(x) = 1 + 2−k . rk ∈(−∞,x]

This function possesses the properties mentioned above and is monotone increasing, but its generalized derivative (in D0 ) has a non-trivial singular component on every interval. Hence for every non-zero element u ∈ D(A) we have u ∈ / D(B), and vice versa. Therefore D(A) ∩D(B) = {θ}. This means that the sum of A and B cannot be defined by adding their values on a common domain. However the quadratic forms of the operators A and B have a common domain W21 (R). Therefore one can define the semi-sum of A and B by adding their quadratic forms. We easily see that the semi-sum 21 (A + B) can also be defined in the sense of Definition 7.

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Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov

The semi-sum of A and B in the sense of quadratic forms coincides with their semi-sum in the sense of Definition 7 and  is a self-adjoint operator with domain d D = u ∈ W21 (R) : (1 + g) dx u ∈ W21 (R) . § 4. A converse of Chernoff ’s theorem The equivalence relation (defined above) on the set of strongly continuous operator-valued functions gives rise to the following question. What conditions on an operator-valued function are sufficient or necessary for the equivalence of this function to a semigroup? The following theorem gives an answer to the first part of this question. Theorem 4 (see [15]). Suppose that F : [0, +∞) → B(X) is continuous in the strong operator topology, F(0) = I and kF(t)kB(X) 6 eαt , t ∈ (0, δ), for some α ∈ R and δ > 0. If the sequence Gn of operator-valued functions   n t , n ∈ N, t ∈ [0, +∞), Gn (t) = F n converges in the strong operator topology uniformly on every interval [0, T ], T > 0, then the limiting operator-valued function is a strongly continuous one-parameter semigroup of operators on X of type ω 6 α. We reproduce a proof of this theorem from [15] for convenience of its discussion in connection with Theorem 5, which generalizes Theorem 4 to the case of non-linear maps. Proof. The functions Gn (t) for all n ∈ N satisfy Gn (0) = I and are continuous maps from [0, +∞) to B(X) in the strong operator topology. The uniform convergence of the sequence {Gn } on every interval [0, T ] in that topology implies that the limiting function G is continuous in it and G(0) = I. Moreover, it follows from the bound on the norm of F that kG(t)kB(X) 6 eαt ∀t > 0. We now prove that G(t + s) = G(t)G(s) for all t, s > 0. 1) Suppose that t > 0 and m ∈ N. Then   n   N m mt t F = F n N provided that n = mN . Since the convergence of {Gn } to G in the strong operator   N m topology for t > 0 implies that for every m ∈ N the sequence F Nt converges to (G(t))m in that topology, we see that the convergence of {Gn } to G as n → ∞ implies that G(mt) = (G(t))m for all t > 0 and m ∈ N. Hence t G m = (G(t))1/m for all t > 0 and m ∈ N. 2) If s = (p/q)t for some q ∈ N, p ∈ Z+ , then s + t = ((p + q)/q)t and by part 1) we have   q   p   q+p t t t G(t) = G , G(s) = G , G(t + s) = G . q q q

Unbounded random operators and Feynman formulae

1145

Hence the function G possesses the semigroup property G(t + s) = G(t)G(s) on the set of rational numbers. 3) Let t, s be arbitrary non-negative numbers and u ∈ X some vector. Since the operator-valued function G is strongly continuous, for every ε > 0 there is a τ > 0 such that ε ε kG(t + ξ)u − G(t)ukX 6 , kG(t + s + ξ)u − G(t + s)ukX 6 2 2 for all ξ ∈ (−τ, τ ) with t − τ > 0. For every s ∈ [0, +∞) one can find a t0 > 0 such that t0 and s are rationally commensurable and |t0 − t| < τ . Therefore G(t0 )G(s) = G(t0 + s). But kG(t)u − G(t0 )uk 6

ε , 2

kG(t + s)u − G(t0 + s)uk 6

ε , 2

whence kG(t + s)u − G(t)G(s)ukX 6 ε and, since the number ε > 0 and the vector u ∈ X are arbitrary, we see that the semigroup property holds for all t, s ∈ R+ .  We notice that Theorem 4 also holds for sequences of iterates of operator-valued functions whose values are non-linear maps from a Banach space X to itself. Indeed, the proof of the theorem does not use the linearity of the operators G(t), t > 0. To state the result on non-linear maps, we define new spaces. Let N (X) be the Banach space of bounded non-linear maps of a Banach space X to itself satisfying A(θ) = θ, with the norm kA(u)kX kAkN (X) = sup . kukX u6=θ A map F : R+ → N (X) is said to be strongly continuous if the map Fu : R+ → X is continuous for every u ∈ X. The following theorem is proved similarly to Theorem 4. Theorem 5. Let F : [0, +∞) → N (X) be a strongly continuous operator-valued function such that F(0) = I,

kF(t)kN (X) 6 eαt ,

t ∈ (0, δ),

for some α ∈ R and δ > 0. If the sequence Gn of operator-valued functions   n t Gn (t) = F , n ∈ N, t ∈ [0, +∞), n converges in the strong operator topology uniformly on every closed interval [0, t], then the limiting operator-valued function is a strongly continuous one-parameter semigroup G of non-linear transformation of X and we have kG(t)kN (X) 6 eαt . For every operator-valued function F : [0, +∞) → B(X) satisfying the hypotheses of Theorem 5, the sequence Gn of operator-valued functions   n t Gn (t) = F , n ∈ N, t ∈ [0, +∞), n

1146

Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov

is compact in the weak operator topology pointwise on R+ . But the convergence of {Gn } to G in the weak operator topology for all t > 0 does not imply that the   N m sequence F Nt converges to (G(t))m in the weak operator topology for every m ∈ N. However, when X is finite-dimensional, the strong and weak operator topologies are equivalent. Therefore if TAn is a sequence of continuous semigroups of linear transformations of a finite-dimensional Banach space X, then the sequence {Gn } of Chernoff iterates of the mean value Z G= TAn dµ(n) N

of the semigroups TAn on a finite-dimensional space X has a unique limit point: the semigroup generated by the averaged operator Z A= An dµ(n). N

Hence, in a finite-dimensional space, the mean value of any random semigroup is Chernoff equivalent to a semigroup whose generator is by Corollary 2 equal to the mean value of the random generators in the Banach space B(X). We now briefly discuss the dependence of the Schr¨odinger group (which is the probabilistic interpolation of quantization of a dynamical system, or group, with random Hamiltonian) on the choice between equivalent classical Hamiltonians in the case when, for example, the only random parameter of the Hamiltonian is the mass distributed according to one and the same law. We assume for simplicity that the mass m is uniformly distributed on the interval [m0 − m0 , m0 + m0 ]. Here m0 characterizes the precision of measurement of this parameter. We put δ = m0 /m0 . For the so-called ‘free non-relativistic’ random classical Hamiltonian H(q, p) = b eqv is e−itHb eqv , t ∈ R, p2 /(2m), the group generated by the mean Hamiltonian H b eqv is equal to where the symbol of H Heqv

1 = 2m0

Z

m0 +m0

m0 −m0

    p2 1 1+δ p2 1 2 p2 2 dm = ln = 1 + δ + o(δ ) . 2m 2m0 2δ 1−δ 2m0 3

Hence the equivalent Hamiltonian has a simple physical interpretation in this example: an inexact estimate for the mass of the system gives rise to an observed motion with effective mass somewhat smaller than m0 . It is less clear how to interpret the solution of the analogous relativistic problem. Here we do not discuss the issue of non-locality of the relativistic Hamiltonian operator, but consider its symbol corresponding to the averaged group. Preserving the notation δ = m0 /m0 and putting x = p/(m0 c), we obtain Heqv =

m0 +m0

c2 2m0

r

p p2 m0 c2 h dm = (1 + δ) 1 + 2δ + x2 2 c 4δ m0 −m0 √ i p m0 c2 2 1 + δ + 1 + 2δ + x2 2 √ − (1 − δ) 1 − 2δ + x + x ln . 4δ 1 − δ + 1 − 2δ + x2

Z

m2 +

Unbounded random operators and Feynman formulae

1147

Although motions with this dependence of the Hamiltonian on x have not been studied, for small δ we have 2

Heqv = m0 c

p

1+

x2

 1+

 3δ 2 x2 δ2 2 + + o(δ ) . 2(1 + x2 )3 4(1 + x2 )2

The expression in square brackets is uniformly bounded with respect to x (along with its derivative) and does not exceed 1 + 45 δ 2 . Nevertheless, this bound for the correction to the theoretical Hamiltonian does not enable us to interpret this correction in terms of the mass of a particle since the mass is also contained under the radical sign. Passing to the non-relativistic limit (m0 c  p) in Heqv , we obtain class Heqv

2

= m0 c



 p2 δ2 2 + o(δ ) + (1 − δ 2 + o(δ 2 )). 1+ 2 2m0

First, this expression differs in the  free-motion part from the equivalent Hamiltonian (p2 /(2m0 )) 1 + 23 δ 2 + o(δ 2 ) obtained above in the non-relativistic case. (This is natural since expansion with respect to a small parameter and averaging with respect to another parameter do not commute in the general case.) Second, the class resulting expression for the effective Hamiltonian Heqv does not enable us to renormalize the mass m0 since this requires us simultaneously to renormalize the speed of light: m0 → m0 (1 + δ 2 ),

c2 → c2 (1 − δ 2 ).

In this case the mean Hamiltonian can be interpreted in physical terms. However, we note that this interpretation differs in the non-relativistic case when the effective mass decreases instead of increasing. Finally, a third version of the same problem is obtained when we start with the weak-relativistic Hamiltonian H WR =

p4 p2 − . 2m 8m2 c2

If the mass parameter is uniformly distributed in the same way as above, then the averaged effective Hamiltonian is given by WR Heqv =

  1+δ p4 1 p2 1 ln − , 2m0 2δ 1−δ 8m20 c2 1 − δ 2

which gives the following asymptotic expression (up to o(δ 2 )) as δ → 0: WR Heqv

  1 2 p4 p2 = 1+ δ − (1 + δ 2 ). 2m0 3 8m20 c2

Thus the influence of averaging of a random Hamiltonian with a random mass parameter may result in different qualitative behavioural effects (increase or decrease) of the effective coefficients of the averaged Hamiltonian.

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Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov

§ 5. Approximation of quantum evolution operators and equilibrium density operators for the harmonic oscillator The following terminology will be used here and in what follows. If an integral operator A : L2 (Q) → L2 (Q) is given by the formula Z Au(x) = K(x, y)u(y) dy, x ∈ Q, Q

then the function K (the ‘integral kernel’ of the operator) is referred to as the matrix of A, and each value of this function as a matrix entry. This terminology is taken from the physics literature and we find it the most natural one. A connection between the choice of quantization of a classical Hamiltonian, the choice of measure in the Feynman–Kac formula, and the choice of approximants for the action functional in Feynman formulae was mentioned in [13]. One considers the measure in the Feynman–Kac integral and the approximant in a Feynman formula and asks how they depend on the choice of quantization. The existence of such a dependence can be seen in the case of a simple harmonic oscillator when one can solve the problem analytically, and explicitly write down the approximation formulae. In [12] we obtained such explicit formulae for the density matrix corresponding to the nth iterate, in the case of Weyl quantization, which enabled us to find the rate of convergence in Chernoff’s theorem. One can also calculate the eτ (x, y) of density matrix for all so-called τ -quantizations when every matrix entry A the operator A given by a classical symbol A(p, q), is related to A(p, q) by an integral transformation, whose (integral) kernel Kτ (p, q | x, y) is called the quantization kernel: Z e Aτ (x, y) = A(q, p)Kτ (q, p | x, y) dq dp, (4) where Kτ (q, p | x, y) =

1 δ(q − τ x − (1 − τ )y) exp[ip(x − y)], 2π

τ ∈ [0, 1].

(5)

When τ = 1/2, the kernel (4) corresponds to the Hermitian (Weyl) quantization. For other values of τ one has non-Hermitian quantizations, which are physically meaningless in the case of observables, but the study of such quantization kernels is useful in finding the dependence of the approximants (in Chernoff’s theorem) on the rule of ordering of non-commuting operators. Consider the ‘one-dimensional harmonic oscillator’, that is, the classical Hamiltonian H(q, p) =

1 2 (q + p2 ). 2

(6)

Every quantization with kernel of the form (4) sends it to the τ -independent Hamiltonian operator 1 H = (q2 + p2 ), (7) 2 where the operators q2 and p2 are defined in accordance with (4), that is, they are ∂2 given by multiplication by x2 and the differentiation operator − ∂x 2 respectively.

Unbounded random operators and Feynman formulae

1149

For this H it is known (see, for example, [15]) that the matrix entries of the operator e−βH , which is referred to as the ‘non-normalized density matrix of the harmonic oscillator’, are of the following form, up to a constant factor:    1 1  2 2 ρ(β; x, y) = √ exp − (x + y ) cosh β − 2xy , 2 sinh β 2π sinh β

(8)

where β > 0 and the quantity 1/β is interpreted as a temperature. In this particular case Chernoff’s theorem states that ρ(β; x, y) = lim ρn (β; x, y) n→∞

(9)

in the following notation: Z ρn (β; x, y) =



β , x, z1 G n

  Z  β β , z1 , z2 · · · G , zn−1 , y dz1 . . . dzn−1 , G n n (10)

Z



Z Kτ (q, p | x, y) exp(−βH(q, p)) dq dp.

G(β, x, y) =

(11)

An exact expression for the matrix of the nth power of the operator with symbol e−βH(q,p) was obtained in [11]: Z

Z Kτ (q1 , p1 | x, z1 ) · · ·

Gn (β, x, y) =

Kτ (qn , pn | zn−1 , y)

  n X H(qk , pk ) dn q dn p dn−1 z × exp −β k=1

=p where

1 π(2β)n An−1

  (−1)n bn 2 An−2 2 2 exp c + b (x + y ) + 2 xy , An−1 An−1



√ n n a2 − 4b2 − a − a2 − 4b2 √ An−1 = , 2n a2 − 4b2 1 β 1 β a = + (τ 2 + (1 − τ )2 ), b = − + τ (1 − τ ), β 2 2β 2 1 2 β (x + y 2 ). c = − (τ 2 x2 + (1 − τ )2 y 2 ) − 2 2β a+

(12)

(13)

These expressions show that explicit formulae are cumbersome even in the relatively simple case of the harmonic oscillator. The formula for the density matrix in the nth approximation follows from (12) in view of the correspondence   β ρn (β; x, y) = Gn ; x, y . (14) n

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Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov

In view of (14), the exact expression (12) for every τ -quantization can be written in the form   Z n β Gτ , x, y = Kτ (q1 , p1 | x, z1 ) · · · Kτ (qn , pn | zn−1 , y) n   n βX H(qk , pk ) dq dp dz × exp − n k=1  2   1 x + y2 β =p exp − coth(β) + 2 n sinh2 (β) 2π sinh(β)   xy β(1 + τ − 2τ 2 )x2 βτ (3 − 2τ )y 2 1 + + + +o . sinh(β) 2n 2n n (15) Let ρ be the density of the probability measure with respect to the Lebesgue measure on [0, 1]. The following theorem gives conditions under which the probabilistic interpolation of the τ -quantizations coincides with the ordinary average. Theorem 6. Suppose that all the operators in the family −Hτ , τ ∈ [0, 1], have a common domain D and each of them generates a contraction semigroup e−tHτ , t > 0. Suppose also that the function [0, 1] 3 τ 7→ Hτ u ∈ X is continuous for every u ∈ D. Then the mean value of the family of semigroups {e−tHτ , t > 0}, τ ∈ [0, 1], with respect to the measure of density ρ is an operator-valued function which is b ρ. Chernoff equivalent to the semigroup generated by the mean Hamiltonian −H Remark. Under the hypotheses of Theorem 6, the generalized mean value of the random semigroup coincides with the semigroup generated by the probabilistic interpolation of the family of Hamiltonians Hτ , τ ∈ [0, 1], by the measure with density Q. Proof. Since the operators Hτ , τ ∈ [0, 1], have a common domain D, the mean bQ value of the Hamiltonian Hρ also has domain D and the value of the operator H on every u ∈ D is given by Z b ρu = b τ uρ(τ ) dτ. H H [0,1]

For every u ∈ D the vector-valued function e−tHτ u, t > 0, has right derivative −Hτ u at the origin, and the integral Z ke−tHτ ukH ρ(τ ) dτ [0,1]

converges uniformly with respect to t > 0. Hence for every u ∈ D the function Z Fµ (t)u = e−tHτ uQ(τ ) dτ [0,1]

has a right derivative at t = 0, and this derivative is equal to Z b ρ u. − Hτ uQ(τ ) dτ ≡ H [0,1]

Unbounded random operators and Feynman formulae

1151

Clearly, for all τ ∈ [0, 1] the semigroups e−tHτ , t > 0, are contraction semigroups. Since the measure Q is non-negative, it follows that all the values of the averaged operator-valued function Fµ are contraction operators. Thus the random Hamiltonian Hτ , τ ∈ [0, 1], satisfies all hypotheses of Theorem 2. This completes the proof.  It thus becomes possible to average the exponential of the operator (15) in the sense of Chernoff equivalence. For a linear quantization (4) we have   Z β n Ghρi , x, y = Kτ (q1 , p1 | x, z1 ) · · · Kτ (qn , pn | zn−1 , y) n   n βX 2 2 × exp − (qk + pk ) dq dp dz n k=1     Z β β = Q(τ1 ) · · · Q(τn )Gτ1 , x, z1 · · · Gτn , zn−1 , y dz1 . . . dzn−1 dτ1 . . . dτn n n  2  2 x +y xy 1 exp − coth(β) + =p 2 sinh(β) 2π sinh(β)    2 2 β β(1 + µ1 − 2µ2 )x2 β(3µ1 − 2µ2 )y 2 1 x +y + + +o . × exp − 2 2n sinh (β) 2n 2n n For Hermitian quantizations we have µ1 = 1/2 and then GnhQi



β , x, y n

 =p

 2  x + y2 exp − coth(β) 2 2π sinh(β)     1 3 1 xy β + − 2µ + o . (16) + + 2 2 n sinh (β) 2 sinh(β) n 1

We now give an example when the spectral properties of the averaged operator differ from those of the operators before averaging in the sense of Definition 7. This example shows that the probabilistic interpolation of a randomization of a family of self-adjoint operators may be a non-selfadjoint operator. It also shows that the summation (integration) method suggested in Definition 7 for unbounded operators is applicable and gives rise to the generator of a semigroup in situations when the summation of operators in the sense of quadratic forms gives a densely defined operator, but even the closure of this operator does not generate a semigroup. Example 2. Consider a maximal symmetric operator L on a Hilbert space H and a sequence of Hamiltonians that converges to L in the topology of strong graph-convergence. Then the following assertion holds (see [19]). Let L be a maximal symmetric operator on a Hilbert space H. Choose a nonnegative normalized purely finitely additive measure µ on N. Let {Ln } be a sequence of self-adjoint operators whose strong graph-limit Γ contains the graph ΓL of the operator L (coincides with it ). If the indices (n− , n+ ) of L are such that n+ = 0, then the sequence of semigroups e−itLn , t > 0, converges in the strong operator topology to the isometric subgroup e−itL , t > 0, uniformly on every closed interval

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Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov

and we have Z

e−itL =

e−itLn dµ(n),

t > 0.

N

But if n− = 0, then the sequence of semigroups e−itLn , t > 0, converges in the ∗ weak operator topology to the contraction semigroup e−itL , t > 0, uniformly on every closed interval and we have Z ∗ e−itL = e−itLn dµ(n), t > 0. N

This assertion deals with a random semigroup of unitary operators ξ : N → Cs (R+ , B(H)) regarded as measurable map from a probability space to the measurable space (Z, Aτs ). The probability space is (N, A, µ), where A is the set 2N of all subsets of N and µ is a non-negative purely finitely additive measure on the measurable space (N, 2N ) (see [21], [22]). Moreover, the Pettis integral Z e−itLn dµ(n) N

is the expectation of the random semigroup. Therefore the following equalities hold in the sense of Definition 7 for any choice of a non-negative normalized purely finitely additive measure µ on N: Z L= Ln dµ(n) for n+ = 0, N

L∗ =

Z Ln dµ(n)

for n+ > 0.

N

Notice that when the strong graph-limit Γ of a sequence {Ln } of self-adjoint operators contains the graph ΓL of T a maximal symmetric operator L (coincides with it), the common domain D = n∈N D(Ln ) need not be dense in H (the vector subspace D may even be trivial). Thus the mean values of random semigroups can be semigroups. There are examples when the mean values of strongly continuous random semigroups are not semigroups, but strongly continuous operator-valued functions (see [23], [19], [24]). § 6. Random operator-valued functions and evolutionary differential equations In the following propositions we find approximations for semigroups of solutions of evolutionary partial differential equations in terms of the averaging of random operator-valued functions generated only by changes of variables. Proposition 1. Suppose that H = L2 (R) and for every ε ∈ R and every v ∈ R there is a family (not a semigroup) Uε,v (t), t > 0, of transformations of H acting by the formula Uε,v (t)u(x) = u(x + vt + εt1/2 ), t > 0.

Unbounded random operators and Feynman formulae

1153

Let µ be a probability measure with density pµ on R such that the function pµ is even and we have Z Z ε2 pµ (ε) dε = D > 0, |ε|3 pµ (ε) dε < ∞. R

R

Then for every v ∈ R the family of averaged transformations Uµv (t), t > 0, is Chernoff equivalent to the semigroup solving the following Cauchy problem for the heat equation: u0t = Du00xx + vu0x ,

t > 0,

x ∈ R,

u|t=+0 = u0 .

(17)

Proof. Clearly, Uε,v (0) = I for all ε ∈ R and, therefore, Uµv (0) = I for any measure µ satisfying the hypotheses of the proposition. It is also clear that kUε,v (t)kB(H) = 1 for all ε ∈ R and t > 0, whence kUµv (t)kB(H) 6 sup kUε,v (t)kB(H) · var(µ) = 1 ε∈R

for any measure µ satisfying the hypotheses of the proposition. Let us prove the differentiability condition. Taking C0∞ (Rd ) for the vector space D, we obtain the following equality for every u ∈ C0∞ . Given any t > 0 and any ε ∈ E, one can find a θ ∈ (0, 1) such that  Uε,v (t)u(x) − u(x) = u(x + vt + εt1/2 ) − u(x) − u0 (x) vt + εt1/2  1  3 1 − u00 (x) v 2 t2 + 2εvt3/2 + ε2 t − u000 x + θ(vt + εt1/2 ) vt + εt1/2 . 2 6 Since the left-hand side is a continuous function of (ε, t, x), so is the right-hand side. Using the hypothesis on the moments of the distribution, we see that for all t > 0 and x ∈ R there is a number θ(ε, t, x) ∈ (0, 1) such that 1 Uµv (t)u(x) − u(x) − u0 (x)vt − u00 (x)Dt 2 Z  3 1 000 = u x + θ(vt + εt1/2 ) vt + εt1/2 pµ (ε) dε. 6 E We have u ∈ C0∞ (R), supξ∈R |u000 (ξ)| = M ∈ (0, +∞) and the expression on the right-hand side is an absolutely integrable function of x by Chebyshev’s inequality. Therefore,

 

1

Uµv (t)u − u − ∆u − v ∂u = o(1) as t → 0.

t ∂x H Notice that the Cauchy problem (17) determines a semigroup V of contractions ∂ of H and its generator is the closure of the operator u → ∆u+v ∂x u with domain D. Hence all the hypotheses of Chernoff’s theorem hold.  The following proposition is proved in a similar way.

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Proposition 2. Suppose that H = L2 (Rd ) and for every ε ∈ Rd and every v ∈ Rd there is a family Uε,v (t), t > 0, of transformations of H acting by the formula Uε,v (t)u(x) = u(x + vt + εt1/2 ),

t > 0.

Let µ be a probability measure with density pµ on Rd such that Z Z Z εpµ dε = 0, εi εj pµ (ε) dε = Di,j ∈ R, |ε|3 pµ (ε) dε < ∞. Rd

Rd

Rd

Then for every v ∈ Rd the family of averaged transformations Uµv (t), t > 0, is Chernoff equivalent to the semigroup solving the following Cauchy problem for the heat equation: u0t = Di,j u00xi xj + vj u0xj ,

x ∈ Rd ,

t > 0,

u|t=+0 = u0 .

Proposition 3. Suppose that H = L2 (R) and for all ε, σ ∈ R and v ∈ R there is a family Uε,σ,v (t), t > 0, of transformations of H acting by the formula Uε,σ,v (t)u(x) = u(x + vt + εt1/2 + σt1/6 ),

t > 0.

Let µ be a probability measure on R with density pµ whose Fourier transform is 2 of the form pbµ (ξ) = e−Dξ . Let ν be a countably additive measure of bounded variation on R which is absolutely continuous with respect to the Lebesgue measure 6 and whose density qν has Fourier transform of the form qbν (ξ) = e−Qξ . Then for every v ∈ Rd the averaged transformation Z Uµ⊗ν ( · ) = Uε,σ,v ( · ) dµ dν v R2

is Chernoff equivalent to the semigroup solving the following Cauchy problem for a sixth-order equation: u0t = Du00xx + vu0x + Qu(6) ,

t > 0,

x ∈ R,

u|t=+0 = u0 .

The proof is similar to that of Proposition 1. By the hypotheses on µ, the function pµ possesses the following properties: Z Z pµ > 0, pµ dε = 1, pµ ε dε = 0, R Z Z R σ 2 pµ (σ) dσ = 2! D > 0, |σ|3 pµ (σ) dσ < ∞. R

R

By the hypotheses on ν, the function qν possesses the following properties: Z qν dσ = 1, R

and all its moments from the first to the fifth vanish, Z Z σ 6 qν (σ) dσ = 6!Q > 0, |σ|7 qν (σ) dσ < ∞. R

R

In particular, qν takes values of both signs and is absolutely integrable.

Unbounded random operators and Feynman formulae

1155

Clearly, Uε,σ,v (0) = I for all (ε, σ) ∈ R2 . Therefore Uµ⊗ν (0) = I for every µ. v µ⊗ν We now estimate the norm kUv (t)kB(H) for t > 0. Take u ∈ L2 (R). Since the Fourier transform F is unitary, we have kUµv (t)ukH = kF[Uµ⊗ν (t)u]kH v for all t > 0. Moreover, since F[Uµ⊗ν (t)u](ξ) = eitvξ Fpµ v



ε



 Fqν

t1/2

σ



t1/6

u b(ξ)

(this is the Fourier transform of a convolution), we have 2

6

F[Uµ⊗ν (t)u](ξ) = eitvξ e−Dξ t e−Qξ t u b(ξ) v and, therefore, kUµv (t)ukH 6 kukH . Arguing as in the proof of Proposition 1, we now see that the map Uvµ⊗ν (t) is right-differentiable at t = 0 on the vector subspace D = C0∞ (R) and all the hypotheses of Chernoff’s theorem (as stated in [9]) hold. Remark. Propositions 1 and 2 show that the semigroup generated by the Cauchy problem for the diffusion equation is a probabilistic interpolation of some family of shift operators. Proposition 3 shows how real-valued quasi-measures can participate in the probabilistic interpolation. Similar arguments show (see [25]) that the probabilistic interpolation of the same family of shifts operators as in Proposition 1, by a complex-valued pseudo-measure coincides with a unitary semigroup of transformations of H and is a randomized quantization of the free-motion Hamiltonian f (q, p) = p2 /2. To represent solutions of evolutionary differential equations in terms of expectations of iterations of random shift operators, one must sometimes average not just over a measure, but over a pseudo-measure (see [26]). By a pseudo-measure on a set E (see [6], [27]–[30]) we mean an additive complex-valued function µ on a certain ring K of subsets of E. Proposition 4. Suppose that H = L2 (R) and for every ε ∈ R there is a family (not a semigroup) Uε (t), t > 0, of transformations of H acting by the formula  Uε (t)u(x) = u x − εt1/2 , t > 0. Let µ be a pseudo-measure on R (that is, an additive complex-valued function on the ring of bounded Lebesgue-measurable subsets of the real line) with density eiπ/4 (−i/(4D))x2 pµ (x) = √ e , 4πD Then the family of averaged transformations Z Uµv (t) = Uε (t) dµ(ε),

x ∈ R.

t > 0,

R

is Chernoff equivalent (and even equal ) to the semigroup solving the following Cauchy problem for the Schr¨ odinger equation: iu0t = Du00xx ,

t > 0,

x ∈ R,

u|t=+0 = u0 .

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Remark. A pseudo-measure µ satisfying the hypotheses of Proposition 2 has bounded variation on every Lebesgue measurable compact subset of R, but the total variation of µ on R is equal to infinity. In [6] such pseudo-measures are referred to as one-dimensional Feynman pseudo-measures, and integration over them is also defined there. Sketch of proof of Proposition 4. If u0 ∈ H, then for every t > 0 we have Z u(t, x) = [Uµt (t)u](x) = u0 (x − t1/2 ε)pµ (ε) dε R   Z y −1/2 = u0 (x − y)t pµ 1/2 dy. t R Hence the function u(t, x) is the convolution of u0 and t−1/2 pµ (y/t1/2 ). Therefore u b(t, ξ) = u b(ξ) · pbµ (t1/2 ξ) and, by the hypotheses on the density pµ , we have 2

u b(t, ξ) = u b(ξ)e−itDξ ,

t > 0.

This proves the proposition. The examples given in Propositions 1–4 show that the expectations of iterates of random one-parameter Feynman–Chernoff families of maps from the finitedimensional Euclidean space Rd to itself are Chernoff equivalent to strongly continuous semigroups of transformations of Lp (Rd ) generated by solutions of partial differential equations. It is also of interest to study such iterates of random one-parameter families of maps of Banach spaces and Banach manifolds (see [22], [31]). The authors are grateful to G. G. Amosov, E. T. Shavgulidze and N. N. Shamarov for fruitful discussions of the problems studied in this paper. Bibliography [1] O. G. Smolyanov, A. G. Tokarev, and A. Truman, “Hamiltonian Feynman path integrals via the Chernoff formula”, J. Math. Phys. 43:10 (2002), 5161–5171. [2] R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics”, Rev. Modern Physics 20 (1948), 367–387. [3] R. P. Feynman, “An operation calculus having applications in quantum electrodynamics”, Physical Rev. (2) 84:1 (1951), 108–128. [4] E. Nelson, “Feynman integrals and the Schr¨ odinger equation”, J. Mathematical Phys. 5:3 (1964), 332–343. [5] P. R. Chernoff, “Note on product formulas for operator semigroups”, J. Functional Analysis 2:2 (1968), 238–242. [6] O. G. Smolyanov and E. T. Shavgulidze, Functional integrals, 2nd ed., rewritten and essentially extended, Lenand, Moscow 2015. (Russian) [7] J. E. Gough, O. O. Obrezkov, and O. G. Smolyanov, “Randomized Hamiltonian Feynman integrals and Shr¨ odinger–Itˆ o stochastic equations”, Izv. Ross. Akad. Nauk Ser. Mat. 69:6 (2005), 3–20; English transl., Izv. Math. 69:6 (2005), 1081–1098. [8] N. A. Sidorova, “The Smolyanov surface measure on trajectories in a Riemannian manifold”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7:3 (2004), 461–471.

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[9] K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Grad. Texts in Math., vol. 194, Springer-Verlag, New York 2000. [10] N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, 2nd extended ed., Nauka, Moscow 1966; English transl. of 1 st ed., vols. I, II, Frederick Ungar, New York 1961, 1963. [11] G. G. Amosov, V. Zh. Sakbaev, and O. G. Smolyanov, “Linear and nonlinear liftings of states of quantum systems”, Russ. J. Math. Phys. 19:4 (2012), 417–427. [12] Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov, “Rate of convergence of Feynman approximations of semigroups generated by the oscillator Hamiltonian”, Teor. Mat. Fiz. 172:1 (2012), 122–137; English transl., Theoret. and Math. Phys. 172:1 (2012), 987–1000. [13] F. A. Berezin, “Non-Wiener functional integrals”, Teor. Mat. Fiz. 6:2 (1971), 194–212; English transl., Theoret. and Math. Phys. 6:2 (1971), 141–155. [14] O. G. Smolyanov, H. v. Weizs¨ acker, and O. Wittich, “Chernoff’s theorem and discrete time approximations of Brownian motion on manifolds”, Potential Anal. 26:1 (2007), 1–29. [15] Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov, “Feynman formulas as a method of averaging random Hamiltonians”, Topics in mathematical physics and analysis, Collection of papers dedicated to the 90th anniversary of Academician Vasilii Sergeevich Vladimirov, Trudy Mat. Inst. Steklova, vol. 285, MAIK, Moscow 2014, pp. 232–241; English transl., Proc. Steklov Inst. Math. 285 (2014), 222–232. [16] Yu. N. Orlov, Foundations of quantization of degenerate dynamical systems, Moscow Institute of Physics and Technology, Moscow 2004. (Russian) [17] K. Yosida, Functional analysis, Grundlehren Math. Wiss., vol. 123, Springer-Verlag, Berlin; Academic Press, New York 1965; Russian transl., Mir, Moscow 1967. [18] T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., vol. 132, Springer-Verlag, New York 1966; Russian transl, Mir, Moscow 1972. [19] V. Zh. Sakbaev, “Cauchy problem for degenerating linear differential equations and averaging of approximating regularizations”, Partial differential equations, MMPS, vol. 43, Russian University of People Friendship, Moscow 2012, pp. 3–172; English transl., J. Math. Sci. (N. Y.) 213:3 (2016), 287–459. [20] V. V. Dodonov, V. I. Man’ko, and V. D. Skarzhinskii, “Ambiguities of variational description of classical systems and the quantization problem”, Quantization, gravitation and group methods in physics, Trudy Fiz. Inst. Akad. Nauk SSSR, vol. 152, Nauka, Moscow 1983, pp. 37–89. (Russian) [21] K. Yosida and E. Hewitt, “Finitely additive measures”, Trans. Amer. Math. Soc. 72 (1952), 46–66. [22] V. Zh. Sakbaev, “Stochastic properties of degenerated quantum systems”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13:1 (2010), 65–85. [23] J. O. Ogun, Yu. N. Orlov, and V. Zh. Sakbaev, “On transformation of the space of initial data for a Cauchy problem with singularities of solutions of blow-up type”, Preprint Keldysh Inst. Applied Math., 2012, 087. (Russian) [24] L. Tartar, “Memory effects and homogenization”, Arch. Rational Mech. Anal. 111:2 (1990), 121–133. [25] L. A. Borisov, Yu. N. Orlolv, and V. Zh. Sakbaev, “Feynman formulae for averaging of semigroups generated by Schr¨ odinger-type operators”, Preprint Keldysh Inst. Applied Math., 2015, 057. (Russian) [26] V. Z. Sakbaev, O. G. Smolyanov, and N. N. Shamarov, “Non-Gaussian Lagrangian Feynman–Kac formulas”, Dokl. Akad. Nauk 457:1 (2014), 28–31; English transl., Dokl. Math. 90:1 (2014), 416–418.

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[27] N. A. Sidorova, O. G. Smolyanov, H. v. Weizs¨ acker, and O. Wittich, “The surface limit of Brownian motion in tubular neighborhoods of an embedded Riemannian manifold”, J. Funct. Anal. 206:2 (2004), 391–413. [28] J. Gough, T. S. Ratiu, and O. G. Smolyanov, “Quantum anomalies and logarithmic derivatives of Feynman pseudo-measures”, Dokl. Akad. Nauk 465:6 (2015), 651–655; English transl., Dokl. Math. 92:3 (2015), 764–768. [29] J. Montaldi and O. G. Smolyanov, “Transformations of measures via their generalized densities”, Russ. J. Math. Phys. 21:3 (2014), 379–385. [30] L. C. Garc´ıa-Naranjo, J. Montaldi, and O. G. Smolyanov, “Transformations of Feynman path integrals and generalized densities of Feynman pseudo-measures”, Dokl. Akad. Nauk 468:4 (2016), 367–371; English transl., Dokl. Math. 93:3 (2016), 282–285. [31] V. Zh. Sakbaev, “On the variational description of the trajectories of averaging quantum dynamical maps”, p-Adic Numbers Ultrametric Anal. Appl. 4:2 (2012), 115–129. Yurii N. Orlov Keldysh Institute of Applied Mathematics, RAS, Moscow E-mail: [email protected] Vsevolod Zh. Sakbaev Moscow Institute of Physics and Thechnology (State University), Dolgoprudnyi, Moscow region Russian University of People Friendship, Moscow E-mail: [email protected] Oleg G. Smolyanov Moscow State University E-mail: [email protected]

Received 29/APR/15 11/FEB/16 Translated by A. V. DOMRIN