On Spectral Theory for Singular Hamiltonians
                   under Left-definiteness Conditions

                                   by

                    Albert Schneider, Univ Dortmund


                AEB350,  3:15pm Tuesday, March 26, 1996

                                Abstract


  On an arbitrary interval I = {a; b} of the real line the Hamiltonian
  system


          +-       -+       +-        -+         +-      -+
          | 0    I  |       | D    D   |         | E    0 |
          |       m |       |  11   12 |         |  11    |
  (1)     |         | y'  + |          | y  =  r |        | y
          |-I    0  |       | D    D   |         |  0   0 |
          |  m      |       |  12   22 |         |        |
          +-       -+       +-       --+         +-      -+

  is considered under the assumptions:


   1. D  (x), D  (x), D  (x), E  (x) are locally integrable on I
       11      12      22      11

   2. D  (x), D  (x), E  (x) are hermitian for x in I, a.e.
       11      22      11

   3. D  (x), -D  (x)  are positive-definite for x in I, a.e.
       11       22

   4. If y is solution of (1) for  r=0  and the usual Hilbert space
      inner product <diag(D  (x),-D  (x))y(x),y(x)> on I equals zero,
                           11      22
      then y = 0,


  and it will be shown, how selfadjoint operators can be generated by
  the system (1) and suitable boundary conditions in a Hilbert space
  setting, defined by the left side of the systems (1) and the boundary
  conditions. One of the main points is, that the spectral theory for
  these operators is developed without using a growth-condition of the
  form

                -s D  (x)  <=  E  (x) <= s D  (x), s > 0
                    11          11          11

  for x in I, a.e., which was originally introduced by H. Weyl in 1910
  in connection with the Sturm-Liouville equation and which has been
  assumed until today in all investigations looking for expansion
  theorems. As one result and application we get a proof of Weyl's
  expansion theorem without the growth-condition now after 80 years.

Requests for preprints and reprints to: schneider@math.uni-dortmund.de
This source can be found at http://www.math.utah.edu/research/