Exact S matrix of the deformed c=1 matrix model.

We consider the c = 1 matrix model deformed by the operator 2M TrΦ −2, which was conjectured by Jevicki and Yoneya to describe a two-dimensional black hole of mass M . We calculate the exact non-perturbative S-matrix and show that all the amplitudes involving an odd number of particles vanish at least to all orders of perturbation theory. We conjecture that these amplitudes vanish non-perturbatively and prove this for the 2n → 1 scattering. For the 2– and 4–particle amplitudes we give some leading terms of the perturbative expansion. 8/93 ⋆ On leave of absence from the Ruder Bošković Institute, Zagreb, Croatia There has been a considerable amount of speculation on the relation between the c = 1 matrix model and two-dimensional stringy black holes [1,2,3,4]. Recently Jevicki and Yoneya [5] made an interesting proposal that a stationary black hole of mass M is described by the large-N Hermitian matrix quantum mechanics with potential U(Φ) = 1 2 Tr(−Φ2 +MΦ−2). The matrix eigenvalues act as free fermions, and their Fermi level μ is set to zero. The deformation of the c = 1 matrix model by the operator TrΦ−2 is uniquely determined by the requirement that it preserve the w∞ symmetry structure [6]. There are further arguments why operators with negative powers of Φ should be identified with “wrongly dressed” Liouville theory operators, of which the black hole mass perturbation is the leading example [5,7–10]. In Ref. [5] some calculations were performed in the deformed matrix model with a number of intriguing results. It was found that 1/ √ M plays the role of the string coupling constant gst, in agreement with string theory in the two-dimensional black hole background. The tree level odd-point functions were found to vanish, which provided one more argument in favor of the black hole analogy. Further studies of the deformed model, including some loop corrections, were performed in [9,10]. In this Letter we calculate the exact non-perturbative S-matrix of the fermion density perturbations in the deformed matrix model. We find that all the odd-point functions vanish at least to all orders in gst. Furthermore, we show that the 2k → 1 amplitudes vanish non-perturbatively and conjecture that this is true for odd-point functions with other kinematical structures. For the 2– and 4–point functions we give a few leading terms of the loop expansion. Our exact solution of the deformed matrix model is based on the powerful method of Moore, Plesser and Ramgoolam [11], who constructed the S-matrix of the c = 1 matrix model in terms of the single-fermion reflection coefficient. Remarkably, in the deformed model the reflection coefficient can also be calculated exactly. The crucial observation is that the single fermion wave function with energy (−ǫ), which satisfies the Schrödinger equation ( d2 dx2 + x − M x2 − 2ǫ ) ψǫ(x) = 0 , (1) is explicitly given by ψǫ(x) = 1 √ 2πx e− iπ 2 (α+ 1 2 ) e−ǫπ/4 |Γ(2 + iǫ 2 + α)| Γ(2α+ 1) Miǫ/2,α(ix ) , (2)