Infinite-dimensional superanalogs of the Mehler formula

Abstract

Remark 2. Take an arbitrary Teichmi~Uer space. Then, by part 1 of the proof, mT(X , tl) generates a semi-metric. The following inequality: rnT(x, y) <_ VT(Z, y) is evident.We find an explicit formula (which generalizes the Mehler formula [1, p. 34]) for the solution of the Cauchy problem for the infinite-dimensional analog of the Schr6dinger equation for the harmonic oscillator in functions on a superspace. The notion of superspace was introduced in [2, 3] (for the infinite-dimensional case) and [4] (for the finite-dimensional case). Note also the papers [5][6][7][8][9][10][11][12][13][14][15][16], which deal with various aspects of finite-and infinite-dimensional superanalysis, and especially Berezin's pioneering work (see [7] and the references therein).In the present paper we use the approach described in [9,11]. Note that the "real" analogs of the problem in question were solved in [17,18]. Superanalogs of classical equations of mathematical physics were also studied in [10-13]. w Superspaces Let A = A0 (3 A1 be a (real) commutative Banach superalgebra with trivial left annihilator [3] and with norm I" I such that labl < [a['lb] for any a, b 9 A. Furthermore, let G = Go@G1 and F be (real)graded Banach spaces, and let E = E0 ~ V ~ W be a real separable Hilbert space such that the subspaces E0 and E1 = V (3 W are orthogonal and V is isomorphic to W. (From now on direct sums are completed in the s norm for Banach spaces and in the t2 norm for Hilbert spaces.) Let us choose orthonormal bases in E0, V, and W. Then the union {e,}i=l of these bases is a basis in E