ISSN 1064–5624, Doklady Mathematics, 2008, Vol. 77, No. 1, pp. 120–123. © Pleiades Publishing, Ltd., 2008. Original Russian Text © M. Gadella, O.G. Smolyanov, 2008, published in Doklady Akademii Nauk, 2008, Vol. 418, No. 6, pp. 727–730.
MATHEMATICS
Feynman Formulas for Particles with Position-Dependent Mass M. Gadellaa and O. G. Smolyanovb Presented by Academician V.V. Kozlov June 28, 2007 Received July 4, 2007
DOI: 10.1134/S1064562408010304
In this paper, we obtain Feynman formulas for solutions to equations describing the diffusion of particles with mass depending on the particle position and to Schrödinger-type equations describing the evolution of quantum particles with similar properties. Such particles (to be more precise, quasi-particles) arise in, e.g., models of semiconductors. Tens of papers studying such models have been published (see [1] and the references therein), but representations of solutions to the arising Schrödinger- and heat-type equations, which go back to Feynman, have not been considered so far. One of the possible reasons is that the traditional application of Feynman’s approach involves integrals with respect to diffusion processes whose transition probabilities have no explicit representation (in terms of elementary functions) in the situation under consideration. In this paper, instead of these transition probabilities, we use their approximations, which can be expressed in terms of elementary functions. Apparently, similar approximations were first applied in [4, 5] to study the diffusion and the quantum evolution of particles of constant mass on Riemannian manifolds. It turns out that the central idea of the approach developed in [4, 5] can also be applied (after appropriate modifications) to the situation considered in this paper. In what follows, we assume that solutions to the Cauchy problems for the equations under examination exist and are unique; thus, we can and shall consider not only solutions to equations but also the corresponding semigroups. Somewhat changing the terminology of [2, 6], we define a real (complex) Schrödinger semigroup as e–tH (respectively, eith), where H is a self-adjoint positive operator on a Hilbert space or the generator of a diffusion process. By a Feynman formula we mean a representation of the solution to the Cauchy problem for the evolution equation as the limit of integrals on Cartesian products
a b
University of Valladolid, Valladolid, Spain Moscow State University, Leninskie gory, Moscow, 119991 Russia e-mail:
[email protected]
of copies of some space as the number of factors tends to infinity. In the case of real Schrödinger semigroups, we usually consider Cartesian powers of the classical configuration space, and in the case of complex semigroups, we consider Cartesian powers of both the configuration and the phase space (it may also be useful (e.g., in BRST quantization) to employ more complicated spaces, but we do not consider them here). As is known, real Schrödinger semigroups can also be represented by the so-called Feynman–Kac formulas, i.e., by using path integrals on the configuration space with respect to probability measures generated by diffusion. A comparison of the Feynman and Feynman–Kac formulas shows that the multiple integrals involved in the Feynman formulas are approximations of integrals with respect to finite-dimensional projections of the probability measures from the Feynman– Kac formulas. Such approximations are useful because they are expressed in terms of elementary functions, while the finite-dimensional projections of measures from the Feynman–Kac formulas may hot have this property. The situation with complex Schrödinger semigroups is quite similar, with the only difference that, instead of probability measures, there arise Feynman pseudomeasures. Moreover, in this case, the Feynman formulas can be considered not only as approximations of Feynman integrals or pseudo-measures defined independently but also as the very definitions of such integrals and pseudo-measures (see [4, 5]). However, in this paper, we do dwell on the relationship between the Feynman formulas and (Gaussian or Feynman) path integrals. The main tool for obtaining Feynman formulas is Chernov’s formula, which is a generalization of wellknown Trotter’s theorem. Apparently, for the first time, Chernov’s formula was applied to obtain Feynmantype formulas in [4, 5]. The further exposition is formal at some places, and some analytic assumptions are omitted.
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1. PRELIMINARIES
1
In this section, we give Chernov’s theorem and describe the relationship between the stochastic Ito equations and the Schrödinger semigroups. For a Banach space X, L(X) denotes the (Banach) space of all continuous linear mappings from X to X endowed with the standard operator norm ||·||. Theorem 1 (Chernov). Suppose that X is a Banach space, F: [0, ∞) L(X) is a strongly continuous mapping with F(0) = Id, where Id is the identity self-mapping of X, and ||F(t)|| ≤ exp{at} for some a ∈ and all t > 0. Suppose also that the restriction of the strong derivative F '(0) of F to some vector subspace of its domain is a closable operator whose closure C is a generator of the strongly continuous semigroup exp{tC}. Then, [F(t/n)]n converges to exp{tC} as n ∞ in the strong operator topology uniformly in t ∈ [0, T] for each T > 0. Remark 1. This statement somewhat differs from the original statement of Chernov (see [5]). Details related to the contents of the remaining part of this section can be found in [3]. Let α: n → L(n) be a function taking values in the set of positive self-adjoint operators, and let ∆α be the differential operator on the space of twice differentiable (real or complex) functions on n defined by (∆αϕ)(x) := trα(x)ϕ''(x) (where ϕ''(x) is the second (Gateaux) derivative of the function ϕ(x) at x ∈ n; the matrix representation of the operator ϕ''(x) is usually called the Hessian of the function ϕ at the point x ∈ n). Definition 1. The operator ∆α is called the Laplace operator generated by the function α (and corresponding to position-dependent mass). For a positive (self-adjoint) operator A,
and D α is some first-order differential operator. This means, in particular, that Feynman formulas for the equation f '(t) = ∆α( f (t)) can be obtained by using results of [4], where Feynman formulas for heat-type equations on Riemannian manifolds were derived. However, it is much simpler to apply the methods of [4] rather than results; we apply them below to obtain Feynman formulas for heat equations on Riemannian manifolds. 2. THE REAL SCHRÖDINGER SEMIGROUPS In this section, we give Feynman formulas for the t∆
α : n →
L(n) is defined by the equality α (x) = each function a: n → n, the equation
α ( x ) . For
f ' ( t ) = ∆ α f ( t ) + ( ( f ( t ) )', a ( · ) )
(1)
Here, f is a mapping from the interval I ⊂ to the space of real-valued functions on n (such a mapping can be identified with a real-valued function on I × n). The symbol ( f (t))' denotes the derivative of a function f (t) on n, while f '(t) denotes the derivative of the function f. If α(x) ≡ 0, then (1) is an equation for the first integrals of the ordinary differential equation (2). Note that ∆α = Dα + where Dα is the Beltrami– Laplace operator with respect to some metric on n [3] 1 Dα ,
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s
t∆
group e α . Since the difference D α – ∆α is a first-order differential operator, it follows that, to obtain Feynman s
s
formulas for the semigroups e
t Dα
s
and e
it D α
, it suffices
to obtain such formulas for the semigroup e
tG α
, where
1 ( G α ϕ ) ( x ) = --- ∆ α ϕ ( x ) + ϕ' ( x )d ( x ) + V ( x )ϕ ( x ). (3) 2 We assume that the functions V(x) and d(x) are differentiable and all of the semigroups under consideration exist. We set X := L2(n) and define functions Fj, where j = 1, 2, by the conditions Fj(0) = Id and, for each t > 0, 1 ( F 1 ( t )ϕ ) ( x ) = -----------------------------------------------n/2 1/2 ( 2πt ) ( detα ( x ) ) ⎧ ( α –1 ( x ) ( z – x ), ( z – x ) ) ⎫ exp ⎨ – -------------------------------------------------------- ⎬ϕ ( z ) dz, 2t ⎩ ⎭ n
∫
×
(2)
s
First, we obtain Feynman formulas for the semi-
(4)
1 ( F 2 ( t )ϕ ) ( x ) = ----------c2 ( t )
is the inverse Kolmogorov equation for the Ito stochastic equation α ( ξ ( t ) )dw ( t ) + a ( ξ ( t ) )dt.
tD
semigroups e α and e α , where D α is the Weyl symmetrization of the operator ∆α.
A denotes
the positive square root of A. The function
dξ ( t ) =
121
⎧ ( α –1 ( x ) ( z – x ), ( z – x ) ) ⎫ exp ⎨ – -------------------------------------------------------- ⎬ϕ ( z ) dz, 2t ⎩ ⎭ n
∫
×
(5)
where c2 ( t ) =
⎧ ( α –1 ( x ) ( z – x ), ( z – x ) ) ⎫ exp ⎨ – -------------------------------------------------------- ⎬dz. 2t ⎩ ⎭ n
∫
(6)
In what follows, instead of (α–1(z)v, v) (where v ∈ n), we write α–1v2.
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Theorem 2. For any ϕ ∈ L2(n), any t > 0, and j = 1, 2, (e
t∆ α /2
t n )ϕ = lim ⎛ F j ⎛ ---⎞ ⎞ ϕ. ⎝ n⎠ ⎠ n ∞⎝
× ϕ ( z + t ( a ( z ) – a ( x ) )dz
∫
= c1 ( t ) e
(7)
Scheme of proof. The required assertion follows from Chernov’s theorem. We only show that, for any smooth function ϕ which decreases sufficiently rapidly together with its derivatives, we have
tV ( x )
n
(10)
⎧ α ( x )(z – x ) exp ⎨ – ---------------------------------2t ⎩ –1
2
( α ( x )a ( x ), ( z – x ) ) ⎫ + --------------------------------------------------- ⎬ 2t ⎭ –1
1 1 --- ( F 1 ( t )ϕ – ϕ ) → --- ∆ α ϕ as t → 0. t 2
⎧ tα –1 ( x ) ( a ( x ) ) 2 ⎫ × exp ⎨ – ------------------------------------ ⎬ϕ ( z )dz. 2 ⎩ ⎭
Let 1 -; c 1 ( t ) = -----------------------------------------------n/2 1/2 ( 2πt ) ( detα ( x ) )
For a function F: [0, β) → L(X), F'(0) denotes its right strong derivative at zero. Theorem 3. For any ϕ ∈ L2(n) and any t > 0,
then, ⎧ α –1 ( x ) ( z – x ) 2 ⎫ ( F 1 ( t )ϕ ) ( x ) = c 1 ( t ) exp ⎨ – ---------------------------------- ⎬ 2t ⎩ ⎭ n
∫
e
(8)
and, therefore, ⎧ α –1 ( x ) ( z – x ) 2 ⎫ 1 1 --- ( F 1 ( t )ϕ – ϕ ) ( x ) = --- c 1 ( t ) exp ⎨ – ---------------------------------- ⎬ t t 2t ⎩ ⎭ n
∫
ϕ'' ( x ) × ⎛ ϕ' ( x ) ( z – x ) + ------------- ( z – x ) ( z – x ) ⎝ 2
(9)
o(t ) 1 t 1 = --- ⎛ --- trα ( x )ϕ'' ( x ) + o ( t )⎞ = --- ∆ α ϕ ( x ) + ---------, ⎝ ⎠ t t 2 2 which completes the proof. Now, we introduce several notations, which we use in the statement of the following theorem. We set (F3(t)ϕ)(x) := ϕ(x + td(x)), (F4(t)ϕ)(x) := etV(x)(x), and F5(t) := F1(t)F3(t)F4(t). Thus,
∫
n
tV ( x )
⎧ α –1 ( x ) ( z – x ) 2 ⎫ exp ⎨ – ---------------------------------- ⎬ 2t ⎩ ⎭
× ϕ ( z + ta ( z ) )dz
∫
= c1 ( t ) e
n
tV ( x )
⎧ α –1 ( x ) ( z – ta ( x ) – x 2 ) ⎫ exp ⎨ – ----------------------------------------------------- ⎬ 2t ⎩ ⎭
(11)
Remark 1. Setting α(x) = const and V(x) ≡ 0 in Theorem 3, we can derive Girsanov’s formula. Thus, Theorem 3 can be regarded as a generalization of Girsanov’s theorem. 3. THE COMPLEX SCHRÖDINGER SEMIGROUPS it∆
itG
For the semigroups e α and e α , results similar to those obtained above are valid. Suppose that, for each t > 0 Φ(t) ∈ L(X), Φ(0) = I, and ( Φ 1 ( t )ϕ ) ( x )
--- + … + o ( z – x ) ⎞ ⎠
( F 5 ( t )ϕ ) ( x ) = c 1 ( t ) e
t n ϕ = lim ⎛ F 5 ⎛ ---⎞ ⎞ ϕ. ⎝ n⎠ ⎠ n ∞⎝
The proof of this theorem uses the equality 1 ( F '5 (0)ϕ)(x) = --- ∆αϕ(x) + ϕ'(x)d(x) + V(x)ϕ(x). 2
ϕ'' ( x ) × ⎛ ϕ ( x ) + ϕ' ( x ) ( z – x ) + ------------- ( z – x ) ( z – x ) ⎝ 2 --- + … + o ( z – x ) ⎞ dz ⎠
tG α
1 = ----------c3 ( t )
⎧ α –1 ( x ) ( z – x ) 2 ⎫ exp ⎨ – i ---------------------------------- ⎬ϕ ( z ) dz, 2t ⎩ ⎭ n
(12)
∫
where c3 ( t ) =
⎧ α –1 ( x ) ( z – x ) 2 ⎫ exp ⎨ – i ---------------------------------- ⎬dz 2t ⎩ ⎭ n
∫
(13)
in both integrals, the standard regularization is used. Theorem 4. For any ϕ ∈ L2(n) and any t > 0, (e
i∆ α t/2
t n )ϕ = lim ⎛ Φ 1 ⎛ ---⎞ ⎞ ϕ. ⎝ n⎠ ⎠ n ∞⎝
(14)
The proof of this theorem is similar to that of Theorem 2. One of the differences is that the function ϕ must be assumed to have compact support. DOKLADY MATHEMATICS
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FEYNMAN FORMULAS FOR PARTICLES WITH POSITION-DEPENDENT MASS
4. REMARKS ON THE CASE OF A DISCONTINUOUS DEPENDENCE OF MASS ON POSITION Suppose that the function d takes only two values, d1 and d2, and the set S of discontinuity points of the function d is a smooth surface S in n. Then, Theorems 2 and 3 remain valid. These theorems give an explicit description (by formulas containing only coefficients and elementary functions) of sequences of probabilities on the space of continuous functions on [0, t] converging to the probability generated by the corresponding diffusion on the same space. We assume that, in the case under consideration, the following additional assumption supplementing Theorem 3 holds. Conjecture 1. For any self-adjoint extension ∆ α of the symmetric operator ∆α (defined on the set of smooth functions vanishing on S together with their derivatives), there exists a sequence {dn} of smooth functions on n taking values in L(X) and such that G d n ∆ α in the sense of strong convergence of the corresponding semigroups. This implies the following proposition, which we also state as a conjecture.
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Conjecture 2. For any self-adjoint extension ∆ α of the operator ∆α, there exists a diffusion on n satisfying the boundary conditions on S determined by this extension and generated by ∆ α . ACKNOWLEDGMENTS O.G. Smolyanov acknowledges the support of the Russian Foundation for Basic Research (project no. 0601-00761a) and the Ministry of education and science of Spain (grant no. SAB2005-0200). REFERENCES 1. M. Gadella, S. Kuru, and J. Negro, Phys. Lett. A 362, 265–268 (2007). 2. B. Simon, Bull. Am. Math. Soc. 7 (3), 447–524 (1982). 3. K. Ito and H. D. McKean, Diffusion Processes and Their Sample Paths (Academic, New York, 1965; Mir, Moscow, 1965). 4. O. G. Smolyanov, H. Weizsäcker, and O. Wittich, Canad. Math. Soc. Conf. Proc. 29, 589–602 (2000). 5. O. G. Smolyanov and A. Truman, Dokl. Math. 68, 194– 198 (2003) [Dokl. Akad. Nauk 392 (2), 174–179 (2003)]. 6. W. Hackenbrock and A. Thalmaier, Stochastische Analyse (Teubner, Stutgart, 1994).
