e i t H notationrecently I saw the notation e i t H , and just wondering how should I interpret it?In my understanding, u ( t , x ) = e i t H u 0 is, for example, a solution to Schrodinger-type equation i ∂ t u = − H u with the initial data u 0 . In case H = Δ, the solution to Schrodinger equation is known to involve the Schrodinger kernel in the integrand. In such case, does e i t H is a short-hand notation for the operator involving the Schrodinger kernel?Or should I interpret e i t H as the Taylor series with H k terms involved? In this case, does the (operator) series converge once applied to the element in the domain of H?Also, I would be very glad to get a reference to read more on this type of operators. Thank you very much!

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e          i      t      H       notationrecently I saw the notation       e          i      t      H      , and just wondering how should I interpret it?In my understanding,   u  (  t  ,  x  )  =      e          i      t      H            u    0   is, for example, a solution to Schrodinger-type equation   i      ∂    t    u  =  −  H  u with the initial data       u    0  . In case   H  =  Δ, the solution to Schrodinger equation is known to involve the Schrodinger kernel in the integrand. In such case, does       e          i      t      H       is a short-hand notation for the operator involving the Schrodinger kernel?Or should I interpret       e          i      t      H       as the Taylor series with       H    k   terms involved? In this case, does the (operator) series converge once applied to the element in the domain of H?Also, I would be very glad to get a reference to read more on this type of operators. Thank you very much!

robegarj · Accepted Answer

Let X be a real or complex Banach space, and let       L    (  X  ) denote the bounded linear operators on       C    0   semigroup on X is a function  T  :  [  0  ,  ∞  )  →      L    (  X  )such that                    (        i        )                                            T        (        0        )        =        I        ,                            (        i        i        )                                            T        (        t        )        T        (                  t          ′                )        =        T        (        t        +                  t          ′                )        ,                            (        i        i        i        )                                                              lim                      t            ↓            0                          T        (        t        )        x        =        x        ,                        ∀        x        ∈        X        .            For any such operator, let       D    (  A  ) denote the set of all   ∈  X for which the following limit exists      lim          h      ↓      0            1    h    (  T  (  h  )  x  −  x  )  ,and let Ax denote this limit. Then   H  :      D    (  A  )  ⊆  X  →  X is a densely-defined linear operator, and we write   T  (  t  )  =      e          t      A       to summarize these properties.If T(t) is unitary for all t&amp;gt;0, then   A  =  i  H, where H is self-adjoint. Then it is customary to write   T  (  t  )  =      e          i      t      H      . This is typical of the time-invariant Schrodinger equation, for example. For such an operator, there is the Borel functional calculus, where f(H)= is defined for any Borel measurable function on R. Using this,       e          i      t      H        =  f  (  H  ) where   f  (  s  )  =      e          i      t      s      .

Reginald Delacruz · Accepted Answer

The underlying Hilbert space has an orthonormal basis consisting of eigenfunctions       u    λ   of H with eigenvalue   λ. Then       e          i      t      H            u    λ    :=      e          i      t      λ            u    λ  . Forcing       e          i      t      H       to be linear and continuous then determines       e          i      t      H        u in general.

e <mrow class="MJX-TeXAtom-ORD"> i t H </mrow> </msup>

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