Hilbert-Schmidt Kernel Research Articles

Quasilocal conservation laws in the quantum Hirota model

The extensivity of the quantum Hirota model’s conservation laws on a 1 + 1 dimensional lattice is considered. This model can be interpreted in terms of an integrable many-body quantum Floquet dynamics. We establish the procedure to generate a continuous family of quasilocal conservation laws from the conserved operators proposed by Faddeev and Volkov. The Hilbert–Schmidt kernel which allows the calculation of inner products of these new conservation laws is explicitly computed. This result has potential applications in quantum quench and transport problems in integrable quantum field theories.

Large deviations for random matrices

Dedicated to Professor K. R. Parthasarathy on the occasion of his 75th birthday Abstract. We prove a large deviation result for a random symmetric n n matrix with independent identically distributed entries to have a few eigen- values of size n. If the spectrum S survives when the matrix is rescaled by a factor of n, it can only be the eigenvalues of a Hilbert-Schmidt kernel k(x;y) on (0;1) (0;1). The rate function for k is I(k) = 1 R h(k(x; y)dxdy where h is the Cramer rate function for the common distribution of the entries that is assumed to have a tail decaying faster than any Gaussian. The large deviation for S is then obtained by contraction.

Special factorization of a non-invertible integral Fredholm operator of the second kind with Hilbert-Schmidt kernel

The problem of the special factorization of a non-invertible integral Fredholm operator of the second kind with Hilbert-Schmidt kernel is considered. Here is the identity operator and is an integral operator: It is proved that is an eigenvalue of of multiplicity if and only if , where the , , , are bounded operators in of a special structure that are invertible from the left and the right, respectively. Bibliography: 7 titles.

Volterra integro-differential equations and infinite systems of ordinary differential equations

We establish a connection between finite-dimensional systems of integro-differential equations with the Hilbert-Schmidt kernel and ordinary differential equations in l 2 (countable systems of differential equations). Such a reduction allows use of results obtained earlier for the countable systems of differential equations in study of integro-differential equations. In particular, it can be employed for study of stability and solutions' constructions for integro-differential equations.

Singular numbers of smooth kernels

In [12] we elaborate the vague principle that the behaviour at infinity of the decreasing sequence of singular numbers sn(K) of a Hilbert–Schmidt kernel K is at least as good as that of the sequence {n−1/qω(n−1;K)}, where ωp is an Lp-modulus of continuity of K and q = p/(p − 1), where 1 ≤ p ≤ 2. Despite the author's effort to justify his study of refinements of the half-century old theorem of Smithies [13], that theorem remains the central result of the subject (viz. that for 0 < a ≤ 1, K∈Lip(a, p) implies that sn(K) = O(n−α−1/q)). For example, Cochran's omnibus theorems [5, 6] that delimit the Schatten classes to which a kernel belongs are based on the blending of ‘smoothness’ conditions and emphasize the pivotal role of the principal corollary of Smithies' theorem (viz. {sn}∈lr if r−1 < α + q−1). Cochran later offered in [7] a very simple derivation of the corollary from a Fourier series theorem of Konyushkov (see [2], vol. II, p. 197), whose proof was, however, at least as intricate as Smithies' demonstration.

Kernels of trace class operators

Let X ⊂ R n X \subset {{\mathbf {R}}^n} and let K K be a trace class operator on L 2 ( X ) {L^2}(X) with corresponding kernel K ( x , y ) ∈ L 2 ( X × X ) K(x,y) \in {L^2}(X \times X) . An integral formula for tr K K , proven by Duflo for continuous kernels, is generalized for arbitrary trace class kernels. This formula is shown to be equivalent to one involving the factorization of K K into a product of Hilbert-Schmidt operators. The formula and its derivation yield two new necessary conditions for traceability of a Hilbert-Schmidt kernel, and these conditions are also shown to be sufficient for positive operators. The proofs make use of the boundedness of the Hardy-Littlewood maximal function on L 2 ( R n ) {L^2}({{\mathbf {R}}^n}) .

Iterative solutions to the nuclear reaction problem

An iterative technique is probably the most efficient and practical way to solve the large sets of integro-differential equations resulting from a CRC analysis of the nuclear reaction problem. In this paper we present the theoretical and practical convergence properties of a new and different type of iterative technique, namely the method of moments. In order to show the power of this method we present a comparison with three other well known iterative methods: the Sasakawa method, the Austern-Sasakawa method, and the method of successive approximation. The dependence of the practical convergence on coupling strength and angular momentum is discussed for the case of inelastic scattering. The method of moments emerges as clearly superior according to both the theoretical and practical convergence criteria. Non-local potentials are shown to introduce very little additional computational difficulty when the iterative technique is used within the framework of the plane-wave expansion method. The method of moments was the only technique capable of guaranteeing convergence when non-local interactions were involved. One merely requires a Hilbert-Schmidt kernel in a finite region of space to guarantee convergence at a rate faster than that of any geometric progression.

Properties of Griffin-Hill-Wheeler spaces

We discuss a method for constructing an explicit orthonormal representation for the collective subspace associated with the generator coordinate method, assuming that the overlap $〈\ensuremath{\alpha}|{\ensuremath{\alpha}}^{\ensuremath{'}}〉$ is a Hilbert-Schmidt kernel in the space of weight functions. We show that the equivalence between the diagonalization of the many-body Hamiltonian in the collective subspace and the solution of the Griffin-Hill-Wheeler equation is a dynamical question which cannot be answered by kinematical considerations alone. The treatment gives a simple picture of well known misbehaviors of the generator coordinate weight functions. An application is made to the Lipkin model.NUCLEAR STRUCTURE Properties of generator coordinate representations. Application to a solvable model.

Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind

We consider approximations{xn}\{ {x_n}\}obtained by moment discretization to (i) the minimalL2{\mathcal {L}_2}-norm solution ofKx=y\mathcal {K}x = ywhereK\mathcal {K}is a Hilbert-Schmidt integral operator onL2{\mathcal {L}_2}, and to (ii) the least squares solution of minimalL2{\mathcal {L}_2}-norm of the same equation whenyis not in the rangeR(K)\mathcal {R}(\mathcal {K})ofK\mathcal {K}. In case (i), ify∈R(K)y \in \mathcal {R}(\mathcal {K}), thenxn→K†y{x_n} \to {\mathcal {K}^\dagger }y, whereK†{\mathcal {K}^\dagger }is the generalized inverse ofK\mathcal {K}, and‖xn‖→∞\left \| {{x_n}} \right \| \to \inftyotherwise. Rates of convergence are given in this case if furtherK†y∈K∗(L2){\mathcal {K}^\dagger }y \in {\mathcal {K}^\ast }({\mathcal {L}_2}), whereK∗{\mathcal {K}^\ast }is the adjoint ofK\mathcal {K}, and the Hilbert-Schmidt kernel ofKK∗\mathcal {K}{\mathcal {K}^\ast }satisfies certain smoothness conditions. In case (ii), ify∈R(K)⊕R(K)⊥y \in \mathcal {R}(\mathcal {K}) \oplus \mathcal {R}{(\mathcal {K})^ \bot }, thenxn→K†y{x_n} \to {\mathcal {K}^\dagger }y, and‖xn‖→∞\left \| {{x_n}} \right \| \to \inftyotherwise. If furtherK†y∈K∗K(L2){\mathcal {K}^\dagger }y \in {\mathcal {K}^\ast }\mathcal {K}({\mathcal {L}_2}), then rates of convergence are given in terms of the smoothness properties of the Hilbert-Schmidt kernel of(KK∗)2{(\mathcal {K}{\mathcal {K}^\ast })^2}. Some of these results are generalized to a class of linear operator equations on abstract Hilbert spaces.

Studies of the three-body problem

In order to test the usefulness of a set of equations derived previously for the three-body problem, we have applied the method of Padé approximants to solve this set. In a model calculation the binding energy of three particles, already known by other methods, is exactly reproduced. Furthermore, the first eigenfunctions for a Hilbert-Schmidt kernel in two variables, which occurs in the set of equations, are determined.

Convergence of the Sasakawa Expansion for the Scattering Amplitude

Sasakawa has rewritten Schr\"odinger's integral equation in such a manner that the inhomogeneous term has the asymptotic behavior of the exact scattering wave function. This paper gives a proof that the iterative solution for the scattering amplitude converges for all local potentials for which the function $\mathrm{rV}(r)$ is absolutely integrable. Under very general conditions the kernel of the Sasakawa equation is a Hilbert-Schmidt kernel. The integral equation is convenient for numerical solution in both the radial representation and the momentum representation.

General Solution of the One-Channel-Scattering Singular Integral Equations

Integral equations for one-channel scattering are written and solved, starting from dispersion relations for the generalized Jost function in the momentum $k$ plane. This method is an alternative to the conventional $\frac{N}{D}$ method, but it allows a simple, physically meaningful generalization to the many-channel case, where dispersion relations and integral equations can be written for a unique generalized Jost function in the complex plane of a suitable variable which uniformizes all the right-hand cuts of the scattering amplitude. Even in the pure elastic-scattering case, a unified treatment is possible, whether the phase shift at infinity is or is not an integral multiple of $\ensuremath{\pi}$. In all cases, our singular integral equations are reduced to a Fredholm-type integral equation with a Hilbert-Schmidt kernel.

Representation of States in a Field Theory with Canonical Variables

We investigate the properties of a functional representation of states for a self-coupled scalar field theory. The assumption is made that all states can be generated by applying functionals of the field at a fixed time ($t=0$) to the vacuum state. It is shown that for the class of models considered the Hamiltonian is uniquely determined by the vacuum functional. The calculation of scalar products between states leads to functional integrals. The measure in this integration over function space is also determined by the vacuum state. Two methods for the evaluation of the functional integrals are discussed. The first one reduces the problem in some simple cases to the solution of an eigenvalue problem for a Hilbert-Schmidt kernel plus a finite number of ordinary integrations. The other one gives a perturbation series.