Smoothing property of the Schrodinger equation
It is well known that the heat diffusion equation
$$ u_t - u_{xx} = 0 ,quad u(0, x ) = f(x) , $$
has the smoothing property.
The question is, how about the imaginary equivalent of it, namely the Schrodinger equation?
$$ i u_t = u_{xx} ,quad u(0, x ) = f(x) . $$
If the initial state $f$ is not derivative at some point, is this property retained by the solution $u$?
pde
asked Nov 30 '18 at 1:43
piepie
735
add a comment |
It is well known that the heat diffusion equation
$$ u_t - u_{xx} = 0 ,quad u(0, x ) = f(x) , $$
has the smoothing property.
The question is, how about the imaginary equivalent of it, namely the Schrodinger equation?
$$ i u_t = u_{xx} ,quad u(0, x ) = f(x) . $$
If the initial state $f$ is not derivative at some point, is this property retained by the solution $u$?
pde
asked Nov 30 '18 at 1:43
piepie
735
See physics.stackexchange.com/questions/281145/…
– obscurans
Nov 30 '18 at 4:02
sometimes it seems, since $f(x)=delta(x)$ corresponds to the fundamental solution, which is analytic
– user254433
Nov 30 '18 at 4:15
add a comment |
It is well known that the heat diffusion equation
$$ u_t - u_{xx} = 0 ,quad u(0, x ) = f(x) , $$
has the smoothing property.
The question is, how about the imaginary equivalent of it, namely the Schrodinger equation?
$$ i u_t = u_{xx} ,quad u(0, x ) = f(x) . $$
If the initial state $f$ is not derivative at some point, is this property retained by the solution $u$?
pde
asked Nov 30 '18 at 1:43
piepie
735
It is well known that the heat diffusion equation
$$ u_t - u_{xx} = 0 ,quad u(0, x ) = f(x) , $$
has the smoothing property.
The question is, how about the imaginary equivalent of it, namely the Schrodinger equation?
$$ i u_t = u_{xx} ,quad u(0, x ) = f(x) . $$
If the initial state $f$ is not derivative at some point, is this property retained by the solution $u$?
pde
pde
asked Nov 30 '18 at 1:43
piepie
735
asked Nov 30 '18 at 1:43
piepie
735
asked Nov 30 '18 at 1:43
piepie
735
asked Nov 30 '18 at 1:43
piepie
735
asked Nov 30 '18 at 1:43
piepie
735
735
See physics.stackexchange.com/questions/281145/…
– obscurans
Nov 30 '18 at 4:02
sometimes it seems, since $f(x)=delta(x)$ corresponds to the fundamental solution, which is analytic
– user254433
Nov 30 '18 at 4:15
add a comment |
See physics.stackexchange.com/questions/281145/…
– obscurans
Nov 30 '18 at 4:02
sometimes it seems, since $f(x)=delta(x)$ corresponds to the fundamental solution, which is analytic
– user254433
Nov 30 '18 at 4:15
See physics.stackexchange.com/questions/281145/…
– obscurans
Nov 30 '18 at 4:02
See physics.stackexchange.com/questions/281145/…
– obscurans
Nov 30 '18 at 4:02
sometimes it seems, since $f(x)=delta(x)$ corresponds to the fundamental solution, which is analytic
– user254433
Nov 30 '18 at 4:15
sometimes it seems, since $f(x)=delta(x)$ corresponds to the fundamental solution, which is analytic
– user254433
Nov 30 '18 at 4:15
add a comment |
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See physics.stackexchange.com/questions/281145/…
– obscurans
Nov 30 '18 at 4:02
sometimes it seems, since $f(x)=delta(x)$ corresponds to the fundamental solution, which is analytic
– user254433
Nov 30 '18 at 4:15