eigenvalues of Laplace-Beltrami on half sphere

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
3
down vote

favorite












Let $ Delta_theta$ denote the Laplace-Beltrami operator on $S^N-1$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^N-1_+$ with zero Dirichlet boundary conditions. Does anyhow know where I can find a reference for these?

thanks
Craig






reference-request ap.analysis-of-pdes sp.spectral-theory harmonic-functions







share|cite|improve this question

























    up vote
    3
    down vote

    favorite












    Let $ Delta_theta$ denote the Laplace-Beltrami operator on $S^N-1$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^N-1_+$ with zero Dirichlet boundary conditions. Does anyhow know where I can find a reference for these?

    thanks
    Craig






    reference-request ap.analysis-of-pdes sp.spectral-theory harmonic-functions







    share|cite|improve this question























      up vote
      3
      down vote

      favorite









      up vote
      3
      down vote

      favorite











      Let $ Delta_theta$ denote the Laplace-Beltrami operator on $S^N-1$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^N-1_+$ with zero Dirichlet boundary conditions. Does anyhow know where I can find a reference for these?

      thanks
      Craig






      reference-request ap.analysis-of-pdes sp.spectral-theory harmonic-functions







      share|cite|improve this question













      Let $ Delta_theta$ denote the Laplace-Beltrami operator on $S^N-1$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^N-1_+$ with zero Dirichlet boundary conditions. Does anyhow know where I can find a reference for these?

      thanks
      Craig






      reference-request ap.analysis-of-pdes sp.spectral-theory harmonic-functions




      reference-request ap.analysis-of-pdes sp.spectral-theory harmonic-functions






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 1 hour ago









      Math604

      36729




      36729




















          2 Answers
          2






          active

          oldest

          votes

















          up vote
          5
          down vote













          Using symmetry, you can extend any Dirichlet eigenfunction on the upper half-sphere to the entire sphere $mathbbS^N-1$. Therefore, the spectrum of the upper hemisphere is a subset of the spectrum of the full sphere. You are searching for the spherical harmonics which vanish on the great circle $x_N equiv 0$. The reference that I've seen that explicitly constructs the $N-1$ dimensional spherical harmonics is the following paper of Frye and Efthimiou: https://arxiv.org/pdf/1205.3548.pdf



          In theory, this reduces your question to a combinatorial problem involving Legendre polynomials, though I haven't solved out the combinatorics explicitly. For the 2-sphere, it seems like the eigenfunctions (and their eigenvalues) you are looking for are the $Y_l^m$ where $m+l$ is odd. From this, you can see the that spectrum is $l(l+1)$ but with less degeneracy than with the full sphere.






          share|cite|improve this answer



























            up vote
            1
            down vote













            Tools for computing eigenvalues on disks in constant-curvature space forms are worked out in Chapter II, section 5 of Chavel's book Eigenvalues in Riemannian Geometry although skimming it I do not see the spectrum itself explicitly written out. Basic idea is separation of variables in polar coordinates.






            share|cite|improve this answer




















              Your Answer




              StackExchange.ifUsing("editor", function ()
              return StackExchange.using("mathjaxEditing", function ()
              StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
              StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
              );
              );
              , "mathjax-editing");

              StackExchange.ready(function()
              var channelOptions =
              tags: "".split(" "),
              id: "504"
              ;
              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',
              convertImagesToLinks: true,
              noModals: false,
              showLowRepImageUploadWarning: true,
              reputationToPostImages: 10,
              bindNavPrevention: true,
              postfix: "",
              noCode: true, onDemand: true,
              discardSelector: ".discard-answer"
              ,immediatelyShowMarkdownHelp:true
              );



              );













               

              draft saved


              draft discarded


















              StackExchange.ready(
              function ()
              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f313170%2feigenvalues-of-laplace-beltrami-on-half-sphere%23new-answer', 'question_page');

              );

              Post as a guest

















              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes








              up vote
              5
              down vote













              Using symmetry, you can extend any Dirichlet eigenfunction on the upper half-sphere to the entire sphere $mathbbS^N-1$. Therefore, the spectrum of the upper hemisphere is a subset of the spectrum of the full sphere. You are searching for the spherical harmonics which vanish on the great circle $x_N equiv 0$. The reference that I've seen that explicitly constructs the $N-1$ dimensional spherical harmonics is the following paper of Frye and Efthimiou: https://arxiv.org/pdf/1205.3548.pdf



              In theory, this reduces your question to a combinatorial problem involving Legendre polynomials, though I haven't solved out the combinatorics explicitly. For the 2-sphere, it seems like the eigenfunctions (and their eigenvalues) you are looking for are the $Y_l^m$ where $m+l$ is odd. From this, you can see the that spectrum is $l(l+1)$ but with less degeneracy than with the full sphere.






              share|cite|improve this answer
























                up vote
                5
                down vote













                Using symmetry, you can extend any Dirichlet eigenfunction on the upper half-sphere to the entire sphere $mathbbS^N-1$. Therefore, the spectrum of the upper hemisphere is a subset of the spectrum of the full sphere. You are searching for the spherical harmonics which vanish on the great circle $x_N equiv 0$. The reference that I've seen that explicitly constructs the $N-1$ dimensional spherical harmonics is the following paper of Frye and Efthimiou: https://arxiv.org/pdf/1205.3548.pdf



                In theory, this reduces your question to a combinatorial problem involving Legendre polynomials, though I haven't solved out the combinatorics explicitly. For the 2-sphere, it seems like the eigenfunctions (and their eigenvalues) you are looking for are the $Y_l^m$ where $m+l$ is odd. From this, you can see the that spectrum is $l(l+1)$ but with less degeneracy than with the full sphere.






                share|cite|improve this answer






















                  up vote
                  5
                  down vote










                  up vote
                  5
                  down vote









                  Using symmetry, you can extend any Dirichlet eigenfunction on the upper half-sphere to the entire sphere $mathbbS^N-1$. Therefore, the spectrum of the upper hemisphere is a subset of the spectrum of the full sphere. You are searching for the spherical harmonics which vanish on the great circle $x_N equiv 0$. The reference that I've seen that explicitly constructs the $N-1$ dimensional spherical harmonics is the following paper of Frye and Efthimiou: https://arxiv.org/pdf/1205.3548.pdf



                  In theory, this reduces your question to a combinatorial problem involving Legendre polynomials, though I haven't solved out the combinatorics explicitly. For the 2-sphere, it seems like the eigenfunctions (and their eigenvalues) you are looking for are the $Y_l^m$ where $m+l$ is odd. From this, you can see the that spectrum is $l(l+1)$ but with less degeneracy than with the full sphere.






                  share|cite|improve this answer












                  Using symmetry, you can extend any Dirichlet eigenfunction on the upper half-sphere to the entire sphere $mathbbS^N-1$. Therefore, the spectrum of the upper hemisphere is a subset of the spectrum of the full sphere. You are searching for the spherical harmonics which vanish on the great circle $x_N equiv 0$. The reference that I've seen that explicitly constructs the $N-1$ dimensional spherical harmonics is the following paper of Frye and Efthimiou: https://arxiv.org/pdf/1205.3548.pdf



                  In theory, this reduces your question to a combinatorial problem involving Legendre polynomials, though I haven't solved out the combinatorics explicitly. For the 2-sphere, it seems like the eigenfunctions (and their eigenvalues) you are looking for are the $Y_l^m$ where $m+l$ is odd. From this, you can see the that spectrum is $l(l+1)$ but with less degeneracy than with the full sphere.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 42 mins ago









                  Gabe K

                  35429




                  35429




















                      up vote
                      1
                      down vote













                      Tools for computing eigenvalues on disks in constant-curvature space forms are worked out in Chapter II, section 5 of Chavel's book Eigenvalues in Riemannian Geometry although skimming it I do not see the spectrum itself explicitly written out. Basic idea is separation of variables in polar coordinates.






                      share|cite|improve this answer
























                        up vote
                        1
                        down vote













                        Tools for computing eigenvalues on disks in constant-curvature space forms are worked out in Chapter II, section 5 of Chavel's book Eigenvalues in Riemannian Geometry although skimming it I do not see the spectrum itself explicitly written out. Basic idea is separation of variables in polar coordinates.






                        share|cite|improve this answer






















                          up vote
                          1
                          down vote










                          up vote
                          1
                          down vote









                          Tools for computing eigenvalues on disks in constant-curvature space forms are worked out in Chapter II, section 5 of Chavel's book Eigenvalues in Riemannian Geometry although skimming it I do not see the spectrum itself explicitly written out. Basic idea is separation of variables in polar coordinates.






                          share|cite|improve this answer












                          Tools for computing eigenvalues on disks in constant-curvature space forms are worked out in Chapter II, section 5 of Chavel's book Eigenvalues in Riemannian Geometry although skimming it I do not see the spectrum itself explicitly written out. Basic idea is separation of variables in polar coordinates.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 46 mins ago









                          Neal

                          450614




                          450614



























                               

                              draft saved


                              draft discarded















































                               


                              draft saved


                              draft discarded














                              StackExchange.ready(
                              function ()
                              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f313170%2feigenvalues-of-laplace-beltrami-on-half-sphere%23new-answer', 'question_page');

                              );

                              Post as a guest
































































                              Comments

                              Post a Comment

                              Popular posts from this blog

                              What does second last employer means? [closed]

                              Image
                              Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote -3 down vote favorite Say, for example I have worked in the following companies, Company A => 2010 - 2011 Company B => 2011 - 2012 Company C => 2012 - Till Date Company D => Have offer Now assume Company D is asking to submit documents from my second last employer. So from the list of companies given, which company would be qualified as second last employer? software-industry employer share | improve this question asked Aug 10 '14 at 5:16 Rajaprabhu Aravindasamy 559 1 6 13 closed as off-topic by Jim G., jmort253 ♦ Aug 11 '14 at 1:58 This question appears to be off-topic. The users who voted to close gave this specific reason: "Questions seeking advice on company-specific regulations, agreements, or policies should be directed to your manager or HR department. Questions that address...
                              Read more

                              Installing NextGIS Connect into QGIS 3?

                              Image
                              Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote 1 down vote favorite Just acquired free version of NEXTGIS and trying to install the plugin for QGIS. It tells me it was not designed for version 3 of QGIS. How do I set up the QGIS-NEXTGIS interface then? qgis qgis-plugins nextgis share | improve this question edited 3 hours ago PolyGeo ♦ 51.9k 17 77 233 asked 4 hours ago IanGS 16 4 New contributor IanGS is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct. add a comment  |  up vote 1 down vote favorite Just acquired free version of NEXTGIS and trying to install the plugin for QGIS. It tells me it was not designed for version 3 of QGIS. How do I set up the QGIS-NEXTGIS interface then? qgis qgis-plugin...
                              Read more

                              One-line joke

                              A one-liner is a joke that is delivered in a single line. A good one-liner is said to be pithy - concise and meaningful. [1] Comedians and actors use this comedic method as part of their act, e.g. Jimmy Carr, Tommy Cooper, Rodney Dangerfield, Norm Macdonald, Ken Dodd, Stewart Francis, Zach Galifianakis, Mitch Hedberg, Anthony Jeselnik, Milton Jones, Shappi Khorsandi, Jay London, Mark Linn-Baker, Demetri Martin, Groucho Marx, Dan Mintz, Emo Philips, Tim Vine, Steven Wright and Henny Youngman and Arnold Schwarzenegger. [ citation needed ] Many fictional characters are also known to deliver one-liners, including James Bond, who usually includes pithy and laconic quips after disposing of a villain. [ citation needed ] Examples "A baby seal walks into a club." "A dyslexic man walks into a bra." (Anonymous) "There are three types of people, those who can count and those who can't." "Venison's dear, isn't it?" (Jimmy Carr) "An escala...
                              Read more