Ttdfjt

How can I watch the 17th (or last, if less) line in files of a folder?

Can I switch to third-person while not in 'town' in Destiny 2?

Why does The Ancient One think differently about Doctor Strange in Endgame than the film Doctor Strange?

How do I find the fastest route from Heathrow to an address in London using all forms of transport?

Why is Boris Johnson visiting only Paris & Berlin if every member of the EU needs to agree on a withdrawal deal?

Gamma GLM - Derive prediction intervals for new x_i

Why doesn't the Falcon-9 first stage use three legs to land?

Most practical knots for hitching a line to an object while keeping the bitter end as tight as possible, without sag?

How would one country purchase another?

LeetCode: Pascal's Triangle C#

What magic extends life or grants immortality?

Why aren't RCS openings an issue for spacecraft heat shields?

Table caption in the middle of the table

What is wrong about this application of Kirchhoffs Current Law?

Factoring the square of this polynomial?

How to compare two different formulations of a problem?

Why did this happen to Thanos's ships at the end of "Avengers: Endgame"?

Concatenation of the result of a function with a mutable default argument in python

If the first law of thermodynamics ensures conservation of energy, why does it allow systems to lose energy?

Co-author responds to email by mistake cc'ing the EiC

How to dismiss intrusive questions from a colleague with whom I don't work?

Justifying the use of directed energy weapons

On the feasibility of space battleships

How does turbine efficiency compare with internal combustion engines if all the turbine power is converted to mechanical energy?

Is the evolution operator well-defined mathematically?

The formal solution of the time-dependent Schrödinger equationWhat exactly does the Hamiltonian operator tell us?How to describe time evolution in relativistic QFT?Prove time-dependent hamiltonian is hermitian from unitarity of time-evolution operatorEvolution operatorWhat's the reasons to use time-ordering operator?Time-dependent quantum mechanical pictureQM: Time evolution with $H = H(t)$Solving the Schrodinger equation with a time-dependent HamiltonianTime evolution operator in QM

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;

8

2

$begingroup$

We know that in order to solve the time-dependent Schrodinger equation $ipartial_t psi = H(t) psi$, we need the evolution operator
$$U(t) = T expleft(-iint_0^t H(t')dt'right)$$
where $T$ is the time ordering operator and the right hand side (RHS) denotes a formal summation.
My question: Does the evolution operator exist in a mathematically rigorous sense? We may deal with imaginary time if necessary.

hilbert-space operators mathematical-physics schroedinger-equation time-evolution

share|cite|improve this question

edited Aug 9 at 7:58

Qmechanic♦

112k1313 gold badges219219 silver badges13341334 bronze badges

asked Aug 9 at 6:57

Andrew YuanAndrew Yuan

17077 bronze badges

$endgroup$

add a comment | 

8

2

$begingroup$

We know that in order to solve the time-dependent Schrodinger equation $ipartial_t psi = H(t) psi$, we need the evolution operator
$$U(t) = T expleft(-iint_0^t H(t')dt'right)$$
where $T$ is the time ordering operator and the right hand side (RHS) denotes a formal summation.
My question: Does the evolution operator exist in a mathematically rigorous sense? We may deal with imaginary time if necessary.

hilbert-space operators mathematical-physics schroedinger-equation time-evolution

share|cite|improve this question

edited Aug 9 at 7:58

Qmechanic♦

112k1313 gold badges219219 silver badges13341334 bronze badges

asked Aug 9 at 6:57

Andrew YuanAndrew Yuan

17077 bronze badges

$endgroup$

add a comment | 

$begingroup$

We know that in order to solve the time-dependent Schrodinger equation $ipartial_t psi = H(t) psi$, we need the evolution operator
$$U(t) = T expleft(-iint_0^t H(t')dt'right)$$
where $T$ is the time ordering operator and the right hand side (RHS) denotes a formal summation.
My question: Does the evolution operator exist in a mathematically rigorous sense? We may deal with imaginary time if necessary.

hilbert-space operators mathematical-physics schroedinger-equation time-evolution

share|cite|improve this question

edited Aug 9 at 7:58

Qmechanic♦

112k1313 gold badges219219 silver badges13341334 bronze badges

asked Aug 9 at 6:57

Andrew YuanAndrew Yuan

17077 bronze badges

$endgroup$

share|cite|improve this question

edited Aug 9 at 7:58

Qmechanic♦

112k1313 gold badges219219 silver badges13341334 bronze badges

asked Aug 9 at 6:57

Andrew YuanAndrew Yuan

17077 bronze badges

share|cite|improve this question

edited Aug 9 at 7:58

Qmechanic♦

112k1313 gold badges219219 silver badges13341334 bronze badges

asked Aug 9 at 6:57

Andrew YuanAndrew Yuan

17077 bronze badges

edited Aug 9 at 7:58

Qmechanic♦

112k1313 gold badges219219 silver badges13341334 bronze badges

edited Aug 9 at 7:58

Qmechanic♦

112k1313 gold badges219219 silver badges13341334 bronze badges

asked Aug 9 at 6:57

Andrew YuanAndrew Yuan

17077 bronze badges

asked Aug 9 at 6:57

Andrew YuanAndrew Yuan

17077 bronze badges

1 Answer
1

active

oldest

votes

10

$begingroup$

If $H(t)$ is selfadjoint and bounded (thus everywhere defined), the theory is standard and quite easy to handle using the uniform operator topology. You can find all required proofs in the first or second volume of Reed and Simon's textbook on mathematical methods. If the operators $H(t)$ are unbounded, the theory is much more difficult also in view of evident problems with domains and it uses the strong operator topology. I am sure that there are classical results by Kato in his celebrated book about the general context ("Perturbation theory of linear operators"), but I suspect there are further more modern results.

share|cite|improve this answer

edited Aug 9 at 15:04

John Rennie

286k4545 gold badges584584 silver badges833833 bronze badges

answered Aug 9 at 11:03

Valter MorettiValter Moretti

38.6k44 gold badges7474 silver badges148148 bronze badges

$endgroup$

add a comment | 

Your Answer

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "151"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fphysics.stackexchange.com%2fquestions%2f495953%2fis-the-evolution-operator-well-defined-mathematically%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

10

$begingroup$

If $H(t)$ is selfadjoint and bounded (thus everywhere defined), the theory is standard and quite easy to handle using the uniform operator topology. You can find all required proofs in the first or second volume of Reed and Simon's textbook on mathematical methods. If the operators $H(t)$ are unbounded, the theory is much more difficult also in view of evident problems with domains and it uses the strong operator topology. I am sure that there are classical results by Kato in his celebrated book about the general context ("Perturbation theory of linear operators"), but I suspect there are further more modern results.

share|cite|improve this answer

edited Aug 9 at 15:04

John Rennie

286k4545 gold badges584584 silver badges833833 bronze badges

answered Aug 9 at 11:03

Valter MorettiValter Moretti

38.6k44 gold badges7474 silver badges148148 bronze badges

$endgroup$

add a comment | 

10

$begingroup$

If $H(t)$ is selfadjoint and bounded (thus everywhere defined), the theory is standard and quite easy to handle using the uniform operator topology. You can find all required proofs in the first or second volume of Reed and Simon's textbook on mathematical methods. If the operators $H(t)$ are unbounded, the theory is much more difficult also in view of evident problems with domains and it uses the strong operator topology. I am sure that there are classical results by Kato in his celebrated book about the general context ("Perturbation theory of linear operators"), but I suspect there are further more modern results.

share|cite|improve this answer

edited Aug 9 at 15:04

John Rennie

286k4545 gold badges584584 silver badges833833 bronze badges

answered Aug 9 at 11:03

Valter MorettiValter Moretti

38.6k44 gold badges7474 silver badges148148 bronze badges

$endgroup$

add a comment | 

$begingroup$

If $H(t)$ is selfadjoint and bounded (thus everywhere defined), the theory is standard and quite easy to handle using the uniform operator topology. You can find all required proofs in the first or second volume of Reed and Simon's textbook on mathematical methods. If the operators $H(t)$ are unbounded, the theory is much more difficult also in view of evident problems with domains and it uses the strong operator topology. I am sure that there are classical results by Kato in his celebrated book about the general context ("Perturbation theory of linear operators"), but I suspect there are further more modern results.

share|cite|improve this answer

edited Aug 9 at 15:04

John Rennie

286k4545 gold badges584584 silver badges833833 bronze badges

answered Aug 9 at 11:03

Valter MorettiValter Moretti

38.6k44 gold badges7474 silver badges148148 bronze badges

$endgroup$

share|cite|improve this answer

edited Aug 9 at 15:04

John Rennie

286k4545 gold badges584584 silver badges833833 bronze badges

answered Aug 9 at 11:03

Valter MorettiValter Moretti

38.6k44 gold badges7474 silver badges148148 bronze badges

edited Aug 9 at 15:04

John Rennie

286k4545 gold badges584584 silver badges833833 bronze badges

edited Aug 9 at 15:04

John Rennie

286k4545 gold badges584584 silver badges833833 bronze badges

answered Aug 9 at 11:03

Valter MorettiValter Moretti

38.6k44 gold badges7474 silver badges148148 bronze badges

answered Aug 9 at 11:03

Valter MorettiValter Moretti

38.6k44 gold badges7474 silver badges148148 bronze badges