zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Solving the dynamical inverse problem for the Schrödinger equation by the boundary control method. (English) Zbl 1001.35121

For an unknown potential q(x), the Schrödinger equation

iu(x,t) t+ 2 u(x,t) x 2 -q(x)u(x,t)=0

with boundary conditions

u(0,t) x=f(t),u(,t)=0,t(0,T)

and initial condition

u(x,0)=0,

gives rise to a response operator (or, in control terms, input-output map)

(R T f)(t)=u f (0,T),

for each control fL 2 (0,T).

The problem considered here is to determine q from R T . If u f (0,t) denotes the solution, for a given control f, at time T and x=0, it is shown that the ‘connection’ operator

C T =(U T ) * U T ,

where U T f=u f (·,T), satisfies C T =i[R T -(R T ) * ]· The spectral data of the operator given by

(ϕ)(x)=-ϕ '' (x)+q(x)ϕ(x)

can be found from C T (and hence from R T ) by the usual variational principle. Finally, the unknown potential q(x) is determined by using the boundary controllability of the associated wave equation. This is based on the fact that the above spectral data completely determine the connecting operator for the associated wave equation.


MSC:
35R30Inverse problems for PDE
35Q40PDEs in connection with quantum mechanics
35J10Schrödinger operator