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

                              Long meetings (6-7 hours a day): Being “babysat” by supervisor

                              Image
                              Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote 31 down vote favorite 4 I hold a PhD in mechanical engineering with a 7 year background in self-guided, self-paced research including 2 years as a graduate level course instructor and a subject matter expert for various engineering projects in a university setting in the United States. For the last 10-11 months, I am working in a research industry in France where my task is to debug C++ spaghetti code written over a period of several years by my current supervisor. It is a team of 2: just I and my supervisor. Diurnally, I am posed with a "bug of the day" to get out of this C++ code. My supervisor generally gives me about 15-30 minutes to solve it and then accosts me about whether or not the task is completed and then rinses and repeats this until the end of the day. Off-late, my supervisor has been holding 6-7 hour long "meetin
                              Read more

                              Is the Concept of Multiple Fantasy Races Scientifically Flawed? [closed]

                              Image
                              Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite Many fantasy worlds have multiple humanoid species (elves, orcs, humans, etc) living in the same world, in addition to that, they often have the ability to interbreed. And I can't help but question their believability... fantasy-races reproduction interspecies-relations share | improve this question edited Aug 9 at 14:46 Michael Kjörling ♦ 21.1k 10 85 161 asked Aug 9 at 13:13 Chlodio 118 6 closed as unclear what you're asking by MichaelK, Gryphon, John, Vincent, Frostfyre Aug 9 at 15:25 Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question. 3 Stack Exchange exist to
                              Read more

                              Confectionery

                              Image
                              Prepared foods made with a lot of sugar or other cabohydrates "Sweetmeat" redirects here. For the racehorse, see Sweetmeat (horse). Not to be confused with Sweetbread. This Kransekake is a traditional Scandinavian baker's confection. Confectionery is the art of making confections , which are food items that are rich in sugar and carbohydrates. Exact definitions are difficult. [1] In general, though, confectionery is divided into two broad and somewhat overlapping categories, bakers' confections and sugar confections. [2] Bakers' confectionery, also called flour confections , includes principally sweet pastries, cakes, and similar baked goods. Sugar confectionery includes candies ( sweets in British English), candied nuts, chocolates, chewing gum, bubble gum, pastillage, and other confections that are made primarily of sugar. In some cases, chocolate confections (confections made of chocolate) are treated as a separate category, as are sugar-free versions of
                              Read more