Square integrable

Here is a complex time which is given by t = t- hl2kT. Methods for evaluating this equation have included path integrals [45], wavepackets [48, 49] and direct evaluation of the trace in square integrable basis sets [ ]. 

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 order for (jc, i) to satisfy equation (2.9), the wave funetion must be square-integrable (also ealled quadratically integrable). Therefore, W(x, /) must go to zero faster than 1 / Z x[ as x approaches ( ) infinity. Likewise, the derivative dW/dx must also go to zero as x approaehes ( ) infinity. 

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. 

Associated with the pole of the S-matrix is a Seigert state, I-Ves, which has purely outgoing boundary conditions and satisfies (with some caveats) the equation, // I res = z les,H being the system Hamiltonian.44 If a square integrable approximation to I res is constructed, then its time evolution, k . (/,), wiH exhibit pure exponential decay after a transient induction period. Of course any L2 state will show quadratic, and hence non-exponential, decay at short times since... 

Note that the condition n = 0 is not allowed since that would imply tp = 0, everywhere, which is not square integrable as required by the Born condition. The ground-state, or zero-point energy... 

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... 

We study the behavior of the solution to Equations (6)-(10), with square integrable gradient in x and y, when 0 and try to obtain an effective problem. 

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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