Results from MathSciNet: Mathematical Reviews on the Web \begin{matrix}c \\ ◯ \end{matrix} Copyright American Mathematical Society 2007

MR0581639 (81h:81198) 81K05 81G10 Schlessinger, L.; Wright, J. A. High-energy bremsstrahlung in an intense laser field. Phys. Rev. A (3) 22 (1980), no. 3, 909–915. Authors’ summary: “The cross section for bremsstrahlung emission and absorption by electrons in an intense laser field has been calculated in the Born approximation for the electron-ion potential. Typical numerical results are presented as a function of the ratio of the electron quiver energy to its energy and the ratio of the bremsstrahlung energy to the electron energy. The intense-field correction factor for the rate of bremsstrahlung emission and absorption for electrons with a Boltzmann distribution of energies has been calculated. Numerical results for the correction factor are presented for the Boltzmann case as a function of the ratio of the electron quiver energy to its thermal energy and the ratio of the bremsstrahlung energy to the thermal energy. For typical laser-fusion parameters, this correction factor which is the ratio of the thermal bremsstrahlung emission rate in the intense laser field to the rate at zero field can be quite significant.”

MR0406256 (53 10047) 81.65 Schlessinger, L.; Payne, G. L. Procedure for finding the scattering solutions of the Schrödinger equation. Phys. Rev. A (3) 10 (1974), 1559–1567. Authors’ summary: “We present a convenient numerical method for the calculation of scattering phase shifts and wave functions. In this procedure the wave function is chosen to be an expansion in a convenient basis set for radial distances within the range of the shortrange interaction. Outside this range, the wave function is a linear combination of the correct asymptotic solutions. Matching these two solutions determines the phase shift and overall normalization of the wave function. This method is an application of the theory first proposed by Bloch. The method is similar in some respects to Rmatrix theory. The computational procedure can also be shown to be equivalent to the Kohn variational principle. In this method the choice of basis states is not as restricted as in previous applications and, in particular, the artificial boundary condition of the usual Rmatrix approach is discarded. The freedom thus obtained in the choice of basis states leads to a rapidly convergent expansion for the wave function and the phase shifts. Furthermore, this freedom enables the method to be easily applied to a wide variety of problems, including the case where there is a long-range Coulomb interaction in addition to short-range interactions. Several numerical examples are presented to illustrate the simplicity, flexibility, accuracy, and rate of convergence of the method.”

MR0344519 (49 9258) 32C40 14B10 Schlessinger, Michael On rigid singularities. Complex analysis, 1972 (Proc. Conf., Rice Univ., Houston, Tex., 1972), Vol. I: Geometry of singularities. Rice Univ. Studies 59 (1973), no. 1, 147–162. Un espace analytique complexe local X est dit rigide si toute famille analytique plate F: V → (T, t_{0}) telle que F^{−1}(t_{0}) ≃ X est triviale (i.e., isomorphe à un produit T × X) au voisinage de t_{0}.

Cet article est une somme de résultats sur les espaces rigides, contenant d’une part un exposé de la théorie des déformations, et expliquant d’autre part comment, si la profondeur de X le long de son lieu singulier Z est assez grande, la théorie des déformations de X se ramène à celle de sa partie lisse X − Z. En particulier, l’auteur utilise cette réduction pour montrer que le quotient de C^{m} par un groupe fini d’automorphismes analytiques G, agissant librement à l’extérieur d’un sous-espace G-invariant W ⊂ C^{m} de codimension au moins 3, est rigide, et pour donner une condition suffisante pour que toute déformation de la singularité au sommet du cône C_{Y} sur une variété projective Y ⊂ P^{n}, de dimension ≥ 2, soit encore un cône, i.e., pour que l’on puisse identifier la théorie des déformations de Y avec celle de C_{Y}. Le dernier paragraphe donne des exemples de singularités qui sont “génériques” en ce sens qu’il n’en existe aucune déformation à un paramètre V → (D, 0) telle que le type analytique de V (t) (t ∈ D^{∗}) soit constant, et qui cependant ne sont pas rigides.

{For more complete bibliographic information about the collection in which this article appears, including the table of contents, see MR0322189 (48 #552a).}

Bernard Teissier MR0339730 (49 4487) 81.49 Payne, G. L.; Schlessinger, L. Variational methods in a harmonic oscillator basis. J. Computational Phys. 13 (1973), 266–287. The authors discuss the advantages of using variational methods (as opposed to numerical integration) for solving the Schrödinger equation to obtain the binding energy, well depth, wave-function, and the potential multiplied by the wave-function for the excited as well as the ground states of a potential. The use of the variational techniques is illustrated by obtaining discrete solutions of the Schrödinger equation for the problem of a particle bound in a Woods-Saxon well of the form

f(r) = {1 + exp[(r − R_{0})/a]}^{−1}.

Upper and lower bounds for the eigenvalue as well as the overlap of the approximate eigenfunction with the true eigenfunction are determined. As basis states the three-dimensional harmonic oscillator functions are used.

H. S. P. Grasser

MR0294701 (45 3769) 32G05 Kas, Arnold; Schlessinger, Michael On the versal deformation of a complex space with an isolated singularity. Math. Ann. 196 (1972), 23–29. This paper represents one of the steps of the progress made recently in the study of deformations of complex analytic spaces with isolated singularities. Let V_{0} be a local complex space, i.e., the set of common zeros of functions f_{1}(z), ⋯, f_{p}(z) holomorphic on C^{n} near 0, two such being identified if the ideals (f_{1}, ⋯, f_{p}) in 𝔒_{C^{n},0} are equal. A deformation of V_{0} is a flat map π: V → T of local complex spaces with π^{−1}(0) = V_{0}. The deformation is versal (or semi-universal) if any other deformation W → S of V_{0} is induced from π by a map S → T such that W = ≈ V ×_{T} S. Assume that V_{0} is a space with an isolated singularity at 0 in C^{n}. Furthermore, V_{0} is a local complete intersection at 0, i.e., there exists a neighborhood 𝒰 of 0 in C^{n} and holomorphic function f_{1}(z), ⋯, f_{p}(z) defined on 𝒰 such that V_{0} ∩ 𝒰 = {z ∈ 𝒰|f_{1}(z) = ⋯ = f_{p}(z) = 0}. It can also be assumed that f_{1}, ⋯, f_{p} generate the ideal of V_{0} at each point of 𝒰. Under these conditions the main result of the paper is that a versal deformation of V_{0} exists. It is worthwhile to point out its construction as follows. Let M be the submodule of (𝔒_{C^{n},0})^{p} generated by the ideal (f_{1}, ⋯, f_{p}) and the ∂f/∂z_{j} = (∂f_{1}/∂z_{j}, ⋯, ∂f_{p}/∂z_{j}) ∈ (𝔒_{C^{n},0})^{p} (j = 1, ⋯, n). Since V_{0} has an isolated singularity at 0, the vector space (𝔒_{C^{n},0})^{p}/M is finite-dimensional. Let P_{1}(z), ⋯, P_{l}(z) be a basis and let V ⊂ 𝒰 × C_{t}^{l} be defined by F_{j}(z, t) = f_{j}(z)+∑ t_{i}P_{i}^{(j)}(z) (j = 1, ⋯, p), P_{i}(z) = (P_{i}^{(1)}(z), ⋯, P_{i}^{(p)}(z)). Then π: V → C_{t}^{l} is the versal flat deformation of V_{0}.

First, the authors clarify the concept of versality. If φ: W → S is any other deformation of V_{0} with S ⊂ C_{s}^{r} and W ⊂ C_{w}^{n} × S defined by p equations of the form G_{j}(w, s) = 0 (j = 1, ⋯, p), G_{j}(w, 0) = f_{j}(w), then the problem asserts the existence of holomorphic mappings C_{z}^{n} × C_{s}^{r} → C_{w}^{n}, C_{s}^{r} → C_{t}, C_{z}^{n} × C_{s}^{r} ∋ (z, s) → H(z, s) ∈ ℳ_{p} (p × p matrices) such that

G(w(z, s), s) = (1_{p} + H(z, s))F (z, t(s)), H(z, 0) = 0.

As usual, the proof is given in two parts. In the first part, through a nice inductive step the authors construct w(z, s), t(s) and H(z, s) as formal power series in s. The second part deals with the convergence of these power series. This is achieved by comparing them with the convergent power series A(s) = (b/(64c)) ∑_{j=1}^{∞}(c^{j}/j^{2})(s_{1} + ⋯ + s_{r})^{j} for suitable c.

The method of comparing with the series A(s) has already been considered by K. Kodaira and D. C. Spencer [Acta Math. 100 (1958), 281–294; MR0112155 (22 #3010)].

G. K. Kiremidjian

MR0292830 (45 1912) 32G05 Schlessinger, Michael Rigidity of quotient singularities. Invent. Math. 14 (1971), 17–26. Let K be the quotient of an abelian variety A under the action of G = Z/2 on A (I ∈ G sends a ∈ A to −a) (characteristic of the base field ≠ 2); K is called a Kummer variety. It is known that the (singular) variety K has more moduli than A in case of surfaces, i.e., dim A = 2. However, as A. Andreotti conjectured, it turns out that the case of dimension ≥ 3 is different: in that case H^{1}(K, θ_{K}) ≅ H^{1}(A, θ_{A}) (Theorem 4), i.e., the first order infinitesimal deformations of K and A coincide. An explanation of this behavior is given in the first section of this paper: For a local scheme X with closed point x, the formal deformation theories of X and of U := X − x are equivalent in case depth_{x}X ≥ 3. This paper sheds new light on the difference in deformation theory between 2- and 3-dimensional varieties; deformations of desingularizations and the aspect of deformations of local versus global schemes seem to be subjects for further study suggested by the methods of this interesting paper.

F. Oort

MR0217093 (36 184) 14.55 Schlessinger, Michael Functors of Artin rings. Trans. Amer. Math. Soc. 130 1968 208–222 Let Λ be a complete noetherian local ring with residue field k, C the category of Artin local Λ-algebras with residue field k, and Ĉ the category of complete noetherian local Λ-algebras R such that R/m^{n} is in C for all n, where m is the maximal ideal of R. A covariant functor F from C to (Sets) such that F (k) ={one point} is called prorepresentable if it is of the form h_{R} for R in Ĉ, where h_{R} is defined by h_{R}(A) = HomĈ(R, A) for A in C.

A. Grothendieck [Fondements de la géométrie algébrique (Extraits du Séminaire Bourbaki, 1957–1962), No. 195, Secrétariat math., Paris, 1962; MR0146040 (26 #3566)] gave a criterion for pro-representability of F, using the techniques of nonflat descent. In the present paper the author gives another simple criterion, depending on the observation of morphisms of certain types in C (small extensions, essential extensions, etc.). For this purpose, he introduces the notion of prorepresentable hull of F. A pair consisting of a ring R in Ĉ and an element ξ of lim_{←}_{n}F (R/m^{n}) is called a pro-representable hull of F if the induced map h_{R} → F is smooth and if the induced map of tangent spaces t_{R} → t_{F} is a bijection, where t_{R} = HomĈ(R, k[ε]) and t_{F} = F (k[ε]), k[ε] being the ring of dual numbers over k. The author’s criterion is stated as follows. Let F be as above, let A′ → A and A′′ → A be morphisms in C, and consider the map (∗) F (A′ ×_{A} A′′) → F (A′) ×_{F (A)} F (A′′), where A′ ×_{A} A′′ is a fibre product of A′ and A′′ over A in C. Then F has a pro-representable hull if and only if F has properties (H_{1}), (H_{2}) and (H_{3}); (H_{1}) (∗) is a surjection if A′ → A is a small extension; (H_{2}) (∗) is a bijection if A = k, A′′ = k[ε]; (H_{3}) dim_{k}(t_{F}) < ∞. F is pro-representable if and only if F has the additional property (H_{4}): (∗) is a bijection if A′ = A′′ and if A′ is a small extension of A.

The author applies this criterion to prove the pro-representability of the Picard functor Pic(X/k), the infinitesimal deformation functor D_{X/k} and the infinitesimal automorphism functor A_{X/k} of a kprescheme X, under suitable finiteness conditions on X.

M. Miyanishi MR0209339 (35 237) 18.20 13.00 Lichtenbaum, S.; Schlessinger, M. The cotangent complex of a morphism. Trans. Amer. Math. Soc. 128 1967 41–70 If B is a commutative A-algebra and M an A-module, the authors define cohomology and homology groups T^{i}(B/A, M) and T_{i}(B/A, M) (i = 0, 1, 2) such that T^{0}(B/A, M) = Der_{A}(B, M) and T_{0}(B/A, M) = Ω_{B/A} ⊗_{B} M. These groups are defined by taking cohomology and homology of a certain three term complex called the co-tangent complex of B over A. The authors show that the groups T^{i} agree with the groups ℰ^{i+1} (i = 0, 1, 2) defined by M. Gerstenhaber for the category of commutative algebras [Ann. of Math. (2) 84 (1966), 1–19; MR0207793 (34 #7608)]. If the base ring is a field, then T^{i} is D. K. Harrison’s ℰ^{i+1} for i = 0, 1 [Trans. Amer. Math. Soc. 104 (1962), 191–204; MR0142607 (26 #176)].

If A → B → C is a sequence of commutative rings, the groups T^{i} fit into a nine term exact sequence which extends the usual exact sequence 0 → Der_{B}(C, M) → Der_{A}(C, M) → Der_{A}(B, M), and similarly for the groups T_{i}. This nine term exact sequence also extends a six term exact sequence of A. Grothendieck [Inst. Hautes Études Sci. Publ. Math. No. 20 (1964); MR0173675 (30 #3885)].

The authors obtain a number of vanishing criteria for the groups T^{i} and T_{i}. For instance, if A is a noetherian ring and B = A/I, then T^{2}(B/A, ·) = 0 if and only if for each prime ideal 𝔮 containing I the ideal I_{𝔮} is generated by an A_{𝔮}-sequence.

If X → Y is a morphism of schemes and if F is a quasi-coherent sheaf of 𝒪_{X}-modules, the definitions in the affine case are globalized and one obtains quasi-coherent sheaves T_{i}(X/Y, F) and, under certain finiteness conditions, T^{i}(X/Y, F). The sheaves T_{i} are employed in a reformulation of the Riemann-Roch-Hirzebruch-Grothendieck theorem.

D. W. Knudsen

MR0183675 (32 1155) 10.05 10.77 Berkovits, S.; Schlessinger, M.; Zierler, N. On mappings for modular arithmetic. II. J. Assoc. Comput. Mach. 12 1965 545–546 The authors examine the same problem as in Part I [#1154 above] with the restriction φ(1) = 1 removed.

R. W. Hamming MR0175520 (30 5704) 81.20 Zierler, Neal; Schlessinger, Michael Boolean embeddings of orthomodular sets and quantum logic. Duke Math. J. 32 1965 251–262 This is one of a series of articles on the axioms of quantum mechanics [N. Zierler, Pacific J. Math. 11 (1961), 1151–1169; MR0140972 (25 #4385); Proc. Amer. Math. Soc. 14 (1963), 346–351; MR0145863 (26 #3391)]. The authors prove the following: Any orthomodular (orthocomplemented and weakly modular) set P has a Boolean extension, but any state for P has no extension to a Boolean algebra unless P has a 2-valued state. This implies that if P is the set of events of nonrelativistic quantum mechanics, i.e., the lattice of closed subspaces of complex separable Hilbert space, then no embedding of P in a deterministic logic exists.

H. Wakita