Square-integrable functions

The formal definition of the Hamiltonian holds at least for infinitely differentiable functions with the boundary condition desired, but this description is insufficient and one must use a more complex construction. For example, for smooth functions, square integrable in 2 simultaneously... 

The function without this fector is of class Q, i.e., normalizable for any finite N, but nuii-iiuuualizable for N = oo. The approximate normalization makes the function square integrable. even for N = oo. Look at the... 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

Firstly, let us formulate an auxiliary statement concerning boundary values for the vector-functions having square integrable divergence (Baiocchi, Capelo, 1984 Temam, 1979). Consider a bounded domain H c i . Introduce the Hilbert space... 

Here is the Sobolev space of functions having square integrable... 

A final example is the concept of QM state. It is often stated that the wave function must be square integrable because the modulus square of the wave function is a probability distribution. States in QM are rays in Hilbert space, which are equivalence classes of wave functions. The equivalence relation between two wave functions is that one wave function is equal to the other multiplied by a complex number. The space of QM states is then a projective space, which by an infinite stereographic projection is isomorphic to a sphere in Hilbert space with any radius, conventionally chosen as one. Hence states can be identified with normalized wave functions as representatives from each equivalence class. This fact is important for the probability interpretation, but it is not a consequence of the probability interpretation. 

One can show (30) that densities are square integrable and thus belong to the Hilbert space I2 (Y) of square integrable functions. This allows one to define... 

Y) Hilbert space of square-integrable complex-yalued functions of 3 real and 2 complex variables. 

Let us define the normalized functions (with respect to square integration over r)... 

In the application of Schrodinger s equation (2.30) to specific physical examples, the requirements that (jc) be continuous, single-valued, and square-integrable restrict the acceptable solutions to an infinite set of specific functions (jc), n = 1, 2, 3,. .., each with a corresponding energy value E . Thus, the energy is quantized, being restricted to certain values. This feature is illustrated in Section 2.5 with the example of a particle in a one-dimensional box. 

In order that the eigenfunctions tp, have physical significance in their application to quantum theory, they are chosen from a special class of functions, namely, those which are continuous, have continuous derivatives, are single-valued, and are square integrable. We refer to functions with these properties as well-behaved functions. Throughout this book we implicitly assume that all functions are well-behaved. 

To describe bound stationary states of the system, the cji s have to be square-normalizable functions. The square-integrability of these functions may be achieved using the following general form of an n-particle correlated Gaussian with the negative exponential of a positive definite quadratic form in 3n variables ... 

A set of nonlinear parameters Aj, in general case, is unique for each function To satisfy the requirement of square integrability of the wave function, each matrix must be positively defined. It imposes certain restrictions on the values that the elements of matrix A may take. To ensure the positive definiteness and to simplify some calclations, it is very convenient to represent matrix A in a Cholesky factored form. 

The Kronecker product with the identity ensures rotational invariance (sphericalness) elliptical Gaussians could be obtained by using a full n x n A matrix. In the former formulation of the basis function, it is difficult to ensure the square integrability of the functions, but this becomes easy in the latter formulation. In this format, all that is required is that the matrix, A, be positive definite. This may be achieved by constructing the matrix from a Cholesky decomposition A), = Later in this work we will use the notation... 

In Eq. (15), 8(rik) is the Dirac delta function which, when integrated with the wave function, gives the value of the wave function at rik = 0. The two terms in Eq. (15) are in reality two limiting forms of the same interaction. The first term is the ordinary dipole-dipole interaction for two dipoles that are not too close to each other. It is the proper form of M S1 to be applied to p, d, and / electrons which are not found near the nucleus. For s electrons, which have a finite probability of being at the nucleus, the first term is clearly inappropriate, since it gives zero contribution at large values of rik and does not hold for small values of rik. From Dirac s relativistic theory of the electron, it is found (4) that the second term in Eq. (15) is the correct form for Si when the electron is close to the nucleus. Thus the contribution toJT S] from s electrons is proportional to the wave function squared at the site of the nucleus and the second term in Eq. (15) is often called the contact term in the hyperfine interaction. 

Proposition 5.5 Let L2(]R O) denote the vector space of square-integrable complex-valued functions on R- . Suppose f and define (using spherical... 

Proposition 7.2 is crucial to our proof in Section 7.2 that the spherical harmonics span the complex scalar product space L (S ) of square-integrable functions on the two-sphere. 

In this section we use the results of Section 7.1 and our knowledge of irreducible representations to show that the spherical harmonic functions span the space of square-integrable functions on the two-sphere. In other... 

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