Adjoint operator

In order to obtain the operator adjoint to M one has to carry out two operations first replace all four M//, M/o, etc., with their adjoint operators and, second, interchange rows and columns. The first step does not introduce any change, since all M//, M/o etc., are self adjoint. This is least evident for the Mff and the Moo- In this case / JV/-uoT/div 1 gradn/dF can be transformed in the following way vq and r/ are assumed to be constants and can be brought before the integration sign. Then, one can write... 

We also require the use of the adjoint of any operator and adjoint functions. Let denote any function satisfying the density boundary conditions and any function satisfying adjoint boundary conditions. Then let M denote the operator adjoint to M such that... 

The last step follows since the slowing-down density, originally and after the change in constants, is zero at both m = 0 and u = Uo] consequently, no specification of q at either limit is required. For this case, then, the operator adjoint to d/du is —d/du. The numerator of the... 

Now perform the general variation of parameters and functions in Elq. (13.96) by applying the delta operator, as used previously in this chapter and, again, choose operators adjoint to 0 and O2 such that... 

Operators that do not obey the above identity are not hermitian. For sueh operators, it is useful to introduee the so-ealled adjoint operator as follows. If for the operator R,... 

These two operators are not Hermitian operators (although Jx and Jy are), but they are adjoints of one another ... 

So far we have not made any assumptions about the properties of the operator A, but we will now assume that A is either self-adjoint or commutes with its hermitean adjoint operator Ht, so that... 

The components of the operator P are hermitian.2 In general, any differential operator Q has a hermitian adjoint Qf, defined by the integral relation... 

An operator is called the adjoint of if they are related as follows ... 

The matrix of L is the transposed conjugate4 of the matrix of L. It is a useful exercise to show that the analog of Eq. (8-19) is true for the adjoint operator ... 

Upon taking the adjoint of Eq. (9-147) we deduce that ( x) anni-hflates the vacuum when operating on it from the right ... 

Since e is the eigenvalue of a hermitian operator, it is real hence, upon taking the hermitian adjoint of Eqs. (9-208) and (9-209) we deduce that... 

Thus Ql —0 does satisfy the correct Heisenberg equation of motion. It should be recalled, incidentally, that the correct definition of the adjoint A of an antilinear operator A is... 

If we assume that the adjoint A of an antilinear operator is defined as in the case of a linear operator by the equation... 

Antiferromagnetic one-dimensional Kronig-Penney potentials, 747 Antiferromagnetic single particle potential, 747 energy band for, 747 Antilinear operator, 687 adjoint of, 688 antihermitian, 688... 

Difference equations with a symmetric matrix are typical in numerical solution of boundary-value problems associated with self-adjoint differential equations of second order. In what follows we will show that the condition Bi = is necessary and sufficient for the operator [yj] be self-adjoint. As can readily be observed, any difference equation of the form... 

Observe that the equality u, Lv)q — (n, Lu) means the self-adjointness of the operator L. 

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